Lecture 13: Differentiation -- Derivatives of Trigonometric Functions
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Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Lecture 13: Differentiation – Derivatives of Trigonometric Functions Derivatives of the Basic Trigonometric Functions Derivative of sin Derivative of cos Using the Chain Rule Derivative of tan Using the Quotient Rule Derivatives the Six Trigonometric Functions Applying the Trig Function Derivative Rules Example 46 – Differentiating with Trig Functions Example 47 – Damped Oscillations Derivatives of the Inverse Trigonometric Functions The arctan Function The arcsin Function Example 48 – Differentiating with Inverse Trig Functions Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 1/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Derivative of sin Derivative of cos Using the Chain Rule Derivative of tan Using the Quotient Rule Derivatives the Six Trigonometric Functions Derivative of sin Recall that in Example 31(c) we guessed that d sin x = cos x dx by considering the graphs of sin and cos. Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 2/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Derivative of sin Derivative of cos Using the Chain Rule Derivative of tan Using the Quotient Rule Derivatives the Six Trigonometric Functions Derivative of sin Recall that in Example 31(c) we guessed that d sin x = cos x dx by considering the graphs of sin and cos. We will now prove this using the definition of the derivative and some basic trigonometric identities. Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 2/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Derivative of sin Derivative of cos Using the Chain Rule Derivative of tan Using the Quotient Rule Derivatives the Six Trigonometric Functions Derivative of sin Recall that in Example 31(c) we guessed that d sin x = cos x dx by considering the graphs of sin and cos. We will now prove this using the definition of the derivative and some basic trigonometric identities. First recall the sum and difference formulas for sin Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 2/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Derivative of sin Derivative of cos Using the Chain Rule Derivative of tan Using the Quotient Rule Derivatives the Six Trigonometric Functions Derivative of sin Recall that in Example 31(c) we guessed that d sin x = cos x dx by considering the graphs of sin and cos. We will now prove this using the definition of the derivative and some basic trigonometric identities. First recall the sum and difference formulas for sin sin (x ± y) Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 2/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Derivative of sin Derivative of cos Using the Chain Rule Derivative of tan Using the Quotient Rule Derivatives the Six Trigonometric Functions Derivative of sin Recall that in Example 31(c) we guessed that d sin x = cos x dx by considering the graphs of sin and cos. We will now prove this using the definition of the derivative and some basic trigonometric identities. First recall the sum and difference formulas for sin sin (x ± y) = sin x cos y ± cos x sin y Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 2/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Derivative of sin Derivative of cos Using the Chain Rule Derivative of tan Using the Quotient Rule Derivatives the Six Trigonometric Functions Derivative of sin Recall that in Example 31(c) we guessed that d sin x = cos x dx by considering the graphs of sin and cos. We will now prove this using the definition of the derivative and some basic trigonometric identities. First recall the sum and difference formulas for sin sin (x ± y) = sin x cos y ± cos x sin y Though we don’t need it right away, the corresponding formula for cos is Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 2/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Derivative of sin Derivative of cos Using the Chain Rule Derivative of tan Using the Quotient Rule Derivatives the Six Trigonometric Functions Derivative of sin Recall that in Example 31(c) we guessed that d sin x = cos x dx by considering the graphs of sin and cos. We will now prove this using the definition of the derivative and some basic trigonometric identities. First recall the sum and difference formulas for sin sin (x ± y) = sin x cos y ± cos x sin y Though we don’t need it right away, the corresponding formula for cos is cos (x ± y) Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 2/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Derivative of sin Derivative of cos Using the Chain Rule Derivative of tan Using the Quotient Rule Derivatives the Six Trigonometric Functions Derivative of sin Recall that in Example 31(c) we guessed that d sin x = cos x dx by considering the graphs of sin and cos. We will now prove this using the definition of the derivative and some basic trigonometric identities. First recall the sum and difference formulas for sin sin (x ± y) = sin x cos y ± cos x sin y Though we don’t need it right away, the corresponding formula for cos is cos (x ± y) = cos x cos y ∓ sin x sin y Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 2/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Derivative of sin Derivative of cos Using the Chain Rule Derivative of tan Using the Quotient Rule Derivatives the Six Trigonometric Functions Derivative of sin – continued With f (x) = sin x, using Formula 3 we have Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 3/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Derivative of sin Derivative of cos Using the Chain Rule Derivative of tan Using the Quotient Rule Derivatives the Six Trigonometric Functions Derivative of sin – continued With f (x) = sin x, using Formula 3 we have f 0 (x) Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 3/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Derivative of sin Derivative of cos Using the Chain Rule Derivative of tan Using the Quotient Rule Derivatives the Six Trigonometric Functions Derivative of sin – continued With f (x) = sin x, using Formula 3 we have f 0 (x) = lim h→0 sin (x + h) − sin x h Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 3/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Derivative of sin Derivative of cos Using the Chain Rule Derivative of tan Using the Quotient Rule Derivatives the Six Trigonometric Functions Derivative of sin – continued With f (x) = sin x, using Formula 3 we have sin (x + h) − sin x h sin x cos h + cos x sin h − sin x = lim h h→0 f 0 (x) = lim h→0 Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 3/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Derivative of sin Derivative of cos Using the Chain Rule Derivative of tan Using the Quotient Rule Derivatives the Six Trigonometric Functions Derivative of sin – continued With f (x) = sin x, using Formula 3 we have f 0 (x) = lim h→0 = lim h→0 = lim h→0 sin (x + h) − sin x h sin x cos h + cos x sin h − sin x h sin x (cos h − 1) + cos x sin h h Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 3/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Derivative of sin Derivative of cos Using the Chain Rule Derivative of tan Using the Quotient Rule Derivatives the Six Trigonometric Functions Derivative of sin – continued With f (x) = sin x, using Formula 3 we have sin (x + h) − sin x h sin x cos h + cos x sin h − sin x = lim h h→0 sin x (cos h − 1) + cos x sin h = lim h h→0 cos h − 1 sin h = sin x lim + cos x lim h h h→0 h→0 f 0 (x) = lim h→0 Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 3/25 Derivative of sin Derivative of cos Using the Chain Rule Derivative of tan Using the Quotient Rule Derivatives the Six Trigonometric Functions Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Derivative of sin – continued With f (x) = sin x, using Formula 3 we have sin (x + h) − sin x h sin x cos h + cos x sin h − sin x = lim h h→0 sin x (cos h − 1) + cos x sin h = lim h h→0 cos h − 1 sin h = sin x lim + cos x lim h h h→0 h→0 f 0 (x) = lim h→0 Recall that using the Squeeze Theorem we proved that lim x→0 sin x x Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 3/25 Derivative of sin Derivative of cos Using the Chain Rule Derivative of tan Using the Quotient Rule Derivatives the Six Trigonometric Functions Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Derivative of sin – continued With f (x) = sin x, using Formula 3 we have sin (x + h) − sin x h sin x cos h + cos x sin h − sin x = lim h h→0 sin x (cos h − 1) + cos x sin h = lim h h→0 cos h − 1 sin h = sin x lim + cos x lim h h h→0 h→0 f 0 (x) = lim h→0 Recall that using the Squeeze Theorem we proved that lim x→0 sin x =1 x Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 3/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Derivative of sin Derivative of cos Using the Chain Rule Derivative of tan Using the Quotient Rule Derivatives the Six Trigonometric Functions Derivative of sin – continued Further, using the same approach as used in Example 13 we can show that lim h→0 cos h − 1 h Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 4/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Derivative of sin Derivative of cos Using the Chain Rule Derivative of tan Using the Quotient Rule Derivatives the Six Trigonometric Functions Derivative of sin – continued Further, using the same approach as used in Example 13 we can show that lim h→0 cos h − 1 =0 h Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 4/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Derivative of sin Derivative of cos Using the Chain Rule Derivative of tan Using the Quotient Rule Derivatives the Six Trigonometric Functions Derivative of sin – continued Further, using the same approach as used in Example 13 we can show that lim h→0 cos h − 1 =0 h Thus we have f 0 (x) = d sin x dx Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 4/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Derivative of sin Derivative of cos Using the Chain Rule Derivative of tan Using the Quotient Rule Derivatives the Six Trigonometric Functions Derivative of sin – continued Further, using the same approach as used in Example 13 we can show that lim h→0 cos h − 1 =0 h Thus we have f 0 (x) = d sin x = sin x (0) + cos x (1) dx Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 4/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Derivative of sin Derivative of cos Using the Chain Rule Derivative of tan Using the Quotient Rule Derivatives the Six Trigonometric Functions Derivative of sin – continued Further, using the same approach as used in Example 13 we can show that lim h→0 cos h − 1 =0 h Thus we have f 0 (x) = d sin x = sin x (0) + cos x (1) = cos x dx Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 4/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Derivative of sin Derivative of cos Using the Chain Rule Derivative of tan Using the Quotient Rule Derivatives the Six Trigonometric Functions Derivative of sin – continued Further, using the same approach as used in Example 13 we can show that lim h→0 cos h − 1 =0 h Thus we have f 0 (x) = d sin x = sin x (0) + cos x (1) = cos x dx as we predicted. Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 4/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Derivative of sin Derivative of cos Using the Chain Rule Derivative of tan Using the Quotient Rule Derivatives the Six Trigonometric Functions Derivative of sin – continued Further, using the same approach as used in Example 13 we can show that lim h→0 cos h − 1 =0 h Thus we have f 0 (x) = d sin x = sin x (0) + cos x (1) = cos x dx as we predicted. You use the sum formula for cos to prove the corresponding differentiation formula for cos x, which is d cos x dx Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 4/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Derivative of sin Derivative of cos Using the Chain Rule Derivative of tan Using the Quotient Rule Derivatives the Six Trigonometric Functions Derivative of sin – continued Further, using the same approach as used in Example 13 we can show that lim h→0 cos h − 1 =0 h Thus we have f 0 (x) = d sin x = sin x (0) + cos x (1) = cos x dx as we predicted. You use the sum formula for cos to prove the corresponding differentiation formula for cos x, which is d cos x = − sin x dx Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 4/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Derivative of sin Derivative of cos Using the Chain Rule Derivative of tan Using the Quotient Rule Derivatives the Six Trigonometric Functions Derivative of cos Using the Chain Rule We can prove the formula for the derivative of cos in a different way. Two basic trigonometric identities are Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 5/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Derivative of sin Derivative of cos Using the Chain Rule Derivative of tan Using the Quotient Rule Derivatives the Six Trigonometric Functions Derivative of cos Using the Chain Rule We can prove the formula for the derivative of cos in a different way. Two basic trigonometric identities are sin π2 − x Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 5/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Derivative of sin Derivative of cos Using the Chain Rule Derivative of tan Using the Quotient Rule Derivatives the Six Trigonometric Functions Derivative of cos Using the Chain Rule We can prove the formula for the derivative of cos in a different way. Two basic trigonometric identities are sin π2 − x = cos x Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 5/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Derivative of sin Derivative of cos Using the Chain Rule Derivative of tan Using the Quotient Rule Derivatives the Six Trigonometric Functions Derivative of cos Using the Chain Rule We can prove the formula for the derivative of cos in a different way. Two basic trigonometric identities are sin π2 − x = cos x cos π2 − x = Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 5/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Derivative of sin Derivative of cos Using the Chain Rule Derivative of tan Using the Quotient Rule Derivatives the Six Trigonometric Functions Derivative of cos Using the Chain Rule We can prove the formula for the derivative of cos in a different way. Two basic trigonometric identities are sin π2 − x = cos x cos π2 − x = sin x Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 5/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Derivative of sin Derivative of cos Using the Chain Rule Derivative of tan Using the Quotient Rule Derivatives the Six Trigonometric Functions Derivative of cos Using the Chain Rule We can prove the formula for the derivative of cos in a different way. Two basic trigonometric identities are sin π2 − x = cos x cos π2 − x = sin x since x and π 2 − x are complementary angles in a right triangle. Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 5/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Derivative of sin Derivative of cos Using the Chain Rule Derivative of tan Using the Quotient Rule Derivatives the Six Trigonometric Functions Derivative of cos Using the Chain Rule We can prove the formula for the derivative of cos in a different way. Two basic trigonometric identities are sin π2 − x = cos x cos π2 − x = sin x since x and π2 − x are complementary angles in a right triangle. Thus, using the Chain Rule gives d cos x dx Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 5/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Derivative of sin Derivative of cos Using the Chain Rule Derivative of tan Using the Quotient Rule Derivatives the Six Trigonometric Functions Derivative of cos Using the Chain Rule We can prove the formula for the derivative of cos in a different way. Two basic trigonometric identities are sin π2 − x = cos x cos π2 − x = sin x since x and π2 − x are complementary angles in a right triangle. Thus, using the Chain Rule gives d d cos x = sin dx dx Clint Lee π 2 −x Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 5/25 Derivative of sin Derivative of cos Using the Chain Rule Derivative of tan Using the Quotient Rule Derivatives the Six Trigonometric Functions Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Derivative of cos Using the Chain Rule We can prove the formula for the derivative of cos in a different way. Two basic trigonometric identities are sin π2 − x = cos x cos π2 − x = sin x since x and π2 − x are complementary angles in a right triangle. Thus, using the Chain Rule gives d d cos x = sin dx dx = cos Clint Lee π 2 π 2 −x −x d dx π 2 −x Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 5/25 Derivative of sin Derivative of cos Using the Chain Rule Derivative of tan Using the Quotient Rule Derivatives the Six Trigonometric Functions Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Derivative of cos Using the Chain Rule We can prove the formula for the derivative of cos in a different way. Two basic trigonometric identities are sin π2 − x = cos x cos π2 − x = sin x since x and π2 − x are complementary angles in a right triangle. Thus, using the Chain Rule gives d d cos x = sin dx dx = cos π 2 π 2 −x = sin x(−1) Clint Lee −x d dx π 2 −x Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 5/25 Derivative of sin Derivative of cos Using the Chain Rule Derivative of tan Using the Quotient Rule Derivatives the Six Trigonometric Functions Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Derivative of cos Using the Chain Rule We can prove the formula for the derivative of cos in a different way. Two basic trigonometric identities are sin π2 − x = cos x cos π2 − x = sin x since x and π2 − x are complementary angles in a right triangle. Thus, using the Chain Rule gives d d cos x = sin dx dx π 2 −x d π −x dx 2 = sin x(−1) = − sin x = cos Clint Lee π 2 −x Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 5/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Derivative of sin Derivative of cos Using the Chain Rule Derivative of tan Using the Quotient Rule Derivatives the Six Trigonometric Functions Derivative of tan Using the Quotient Rule Recall that tan x Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 6/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Derivative of sin Derivative of cos Using the Chain Rule Derivative of tan Using the Quotient Rule Derivatives the Six Trigonometric Functions Derivative of tan Using the Quotient Rule Recall that tan x = Clint Lee sin x cos x Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 6/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Derivative of sin Derivative of cos Using the Chain Rule Derivative of tan Using the Quotient Rule Derivatives the Six Trigonometric Functions Derivative of tan Using the Quotient Rule Recall that tan x = sin x cos x Thus, using the Quotient Rule gives d tan x dx Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 6/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Derivative of sin Derivative of cos Using the Chain Rule Derivative of tan Using the Quotient Rule Derivatives the Six Trigonometric Functions Derivative of tan Using the Quotient Rule Recall that tan x = sin x cos x Thus, using the Quotient Rule gives d d sin x tan x = dx dx cos x Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 6/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Derivative of sin Derivative of cos Using the Chain Rule Derivative of tan Using the Quotient Rule Derivatives the Six Trigonometric Functions Derivative of tan Using the Quotient Rule Recall that tan x = sin x cos x Thus, using the Quotient Rule gives d d sin x tan x = dx dx cos x d d sin x cos x − sin x cos x dx dx = (cos x)2 Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 6/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Derivative of sin Derivative of cos Using the Chain Rule Derivative of tan Using the Quotient Rule Derivatives the Six Trigonometric Functions Derivative of tan Using the Quotient Rule Recall that tan x = sin x cos x Thus, using the Quotient Rule gives d d sin x tan x = dx dx cos x d d sin x cos x − sin x cos x dx dx = (cos x)2 (cos x) cos x − sin x (− sin x) = cos2 x Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 6/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Derivative of sin Derivative of cos Using the Chain Rule Derivative of tan Using the Quotient Rule Derivatives the Six Trigonometric Functions Derivative of tan Using the Quotient Rule Recall that tan x = sin x cos x Thus, using the Quotient Rule gives d d sin x tan x = dx dx cos x d d sin x cos x − sin x cos x dx dx = (cos x)2 (cos x) cos x − sin x (− sin x) = cos2 x 2 cos x + sin2 x = cos2 x Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 6/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Derivative of sin Derivative of cos Using the Chain Rule Derivative of tan Using the Quotient Rule Derivatives the Six Trigonometric Functions Derivative of tan Using the Quotient Rule Recall that tan x = sin x cos x Thus, using the Quotient Rule gives d d sin x tan x = dx dx cos x d d sin x cos x − sin x cos x dx dx = (cos x)2 (cos x) cos x − sin x (− sin x) = cos2 x 2 1 cos x + sin2 x = = cos2 x cos2 x Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 6/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Derivative of sin Derivative of cos Using the Chain Rule Derivative of tan Using the Quotient Rule Derivatives the Six Trigonometric Functions Derivative of tan Using the Quotient Rule Recall that tan x = sin x cos x Thus, using the Quotient Rule gives d d sin x tan x = dx dx cos x d d sin x cos x − sin x cos x dx dx = (cos x)2 (cos x) cos x − sin x (− sin x) = cos2 x 2 1 cos x + sin2 x = = cos2 x cos2 x 2 1 = cos x Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 6/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Derivative of sin Derivative of cos Using the Chain Rule Derivative of tan Using the Quotient Rule Derivatives the Six Trigonometric Functions Derivative of tan Using the Quotient Rule Recall that tan x = sin x cos x Thus, using the Quotient Rule gives d d sin x tan x = dx dx cos x d d sin x cos x − sin x cos x dx dx = (cos x)2 (cos x) cos x − sin x (− sin x) = cos2 x 2 1 cos x + sin2 x = = cos2 x cos2 x 2 1 = = sec2 x cos x Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 6/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Derivative of sin Derivative of cos Using the Chain Rule Derivative of tan Using the Quotient Rule Derivatives the Six Trigonometric Functions Derivative of tan Using the Quotient Rule – continued An alternative way to simplify the previous expression is d tan x dx Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 7/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Derivative of sin Derivative of cos Using the Chain Rule Derivative of tan Using the Quotient Rule Derivatives the Six Trigonometric Functions Derivative of tan Using the Quotient Rule – continued An alternative way to simplify the previous expression is d cos2 x + sin2 x tan x = dx cos2 x Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 7/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Derivative of sin Derivative of cos Using the Chain Rule Derivative of tan Using the Quotient Rule Derivatives the Six Trigonometric Functions Derivative of tan Using the Quotient Rule – continued An alternative way to simplify the previous expression is cos2 x sin2 x d cos2 x + sin2 x = + tan x = 2 2 dx cos x cos x cos2 x Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 7/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Derivative of sin Derivative of cos Using the Chain Rule Derivative of tan Using the Quotient Rule Derivatives the Six Trigonometric Functions Derivative of tan Using the Quotient Rule – continued An alternative way to simplify the previous expression is cos2 x sin2 x d cos2 x + sin2 x = + tan x = 2 2 dx cos x cos x cos2 x 2 sin x = 1+ cos x Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 7/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Derivative of sin Derivative of cos Using the Chain Rule Derivative of tan Using the Quotient Rule Derivatives the Six Trigonometric Functions Derivative of tan Using the Quotient Rule – continued An alternative way to simplify the previous expression is cos2 x sin2 x d cos2 x + sin2 x = + tan x = 2 2 dx cos x cos x cos2 x 2 sin x = 1 + tan2 x = 1+ cos x Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 7/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Derivative of sin Derivative of cos Using the Chain Rule Derivative of tan Using the Quotient Rule Derivatives the Six Trigonometric Functions Derivative of tan Using the Quotient Rule – continued An alternative way to simplify the previous expression is cos2 x sin2 x d cos2 x + sin2 x = + tan x = 2 2 dx cos x cos x cos2 x 2 sin x = 1 + tan2 x = 1+ cos x So that d tan x dx Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 7/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Derivative of sin Derivative of cos Using the Chain Rule Derivative of tan Using the Quotient Rule Derivatives the Six Trigonometric Functions Derivative of tan Using the Quotient Rule – continued An alternative way to simplify the previous expression is cos2 x sin2 x d cos2 x + sin2 x = + tan x = 2 2 dx cos x cos x cos2 x 2 sin x = 1 + tan2 x = 1+ cos x So that d tan x = sec2 x dx Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 7/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Derivative of sin Derivative of cos Using the Chain Rule Derivative of tan Using the Quotient Rule Derivatives the Six Trigonometric Functions Derivative of tan Using the Quotient Rule – continued An alternative way to simplify the previous expression is cos2 x sin2 x d cos2 x + sin2 x = + tan x = 2 2 dx cos x cos x cos2 x 2 sin x = 1 + tan2 x = 1+ cos x So that d tan x = sec2 x = 1 + tan2 x dx Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 7/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Derivative of sin Derivative of cos Using the Chain Rule Derivative of tan Using the Quotient Rule Derivatives the Six Trigonometric Functions Derivative of tan Using the Quotient Rule – continued An alternative way to simplify the previous expression is cos2 x sin2 x d cos2 x + sin2 x = + tan x = 2 2 dx cos x cos x cos2 x 2 sin x = 1 + tan2 x = 1+ cos x So that d tan x = sec2 x = 1 + tan2 x dx The equality above can also be proved using the Pythagorean identity 1 + tan2 x Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 7/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Derivative of sin Derivative of cos Using the Chain Rule Derivative of tan Using the Quotient Rule Derivatives the Six Trigonometric Functions Derivative of tan Using the Quotient Rule – continued An alternative way to simplify the previous expression is cos2 x sin2 x d cos2 x + sin2 x = + tan x = 2 2 dx cos x cos x cos2 x 2 sin x = 1 + tan2 x = 1+ cos x So that d tan x = sec2 x = 1 + tan2 x dx The equality above can also be proved using the Pythagorean identity 1 + tan2 x = sec2 x Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 7/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Derivative of sin Derivative of cos Using the Chain Rule Derivative of tan Using the Quotient Rule Derivatives the Six Trigonometric Functions Derivative of tan Using the Quotient Rule – continued An alternative way to simplify the previous expression is cos2 x sin2 x d cos2 x + sin2 x = + tan x = 2 2 dx cos x cos x cos2 x 2 sin x = 1 + tan2 x = 1+ cos x So that d tan x = sec2 x = 1 + tan2 x dx The equality above can also be proved using the Pythagorean identity 1 + tan2 x = sec2 x Most text books use the sec2 x formula for the derivative of tan x, but Maple and other symbolic differentiating programs use the 1 + tan2 x formula. Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 7/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Derivative of sin Derivative of cos Using the Chain Rule Derivative of tan Using the Quotient Rule Derivatives the Six Trigonometric Functions Derivatives the Six Trigonometric Functions Using basic differentiation rules as in the derivation of the derivative formula for tan we can find derivative formulas for all of the other trigonometric functions. Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 8/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Derivative of sin Derivative of cos Using the Chain Rule Derivative of tan Using the Quotient Rule Derivatives the Six Trigonometric Functions Derivatives the Six Trigonometric Functions Using basic differentiation rules as in the derivation of the derivative formula for tan we can find derivative formulas for all of the other trigonometric functions. Also, recall that when we derived the General Rule for the Exponential Function we stated that we would give all derivative formulas in a general form using the Chain Rule. In this form we introduce an intermediate variable u assumed to represent some function of x. With this assumption the derivative rules for all six basic trigonometric functions are: Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 8/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Derivative of sin Derivative of cos Using the Chain Rule Derivative of tan Using the Quotient Rule Derivatives the Six Trigonometric Functions Derivatives the Six Trigonometric Functions Using basic differentiation rules as in the derivation of the derivative formula for tan we can find derivative formulas for all of the other trigonometric functions. Also, recall that when we derived the General Rule for the Exponential Function we stated that we would give all derivative formulas in a general form using the Chain Rule. In this form we introduce an intermediate variable u assumed to represent some function of x. With this assumption the derivative rules for all six basic trigonometric functions are: d sin u dx Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 8/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Derivative of sin Derivative of cos Using the Chain Rule Derivative of tan Using the Quotient Rule Derivatives the Six Trigonometric Functions Derivatives the Six Trigonometric Functions Using basic differentiation rules as in the derivation of the derivative formula for tan we can find derivative formulas for all of the other trigonometric functions. Also, recall that when we derived the General Rule for the Exponential Function we stated that we would give all derivative formulas in a general form using the Chain Rule. In this form we introduce an intermediate variable u assumed to represent some function of x. With this assumption the derivative rules for all six basic trigonometric functions are: d du sin u = cos u dx dx Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 8/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Derivative of sin Derivative of cos Using the Chain Rule Derivative of tan Using the Quotient Rule Derivatives the Six Trigonometric Functions Derivatives the Six Trigonometric Functions Using basic differentiation rules as in the derivation of the derivative formula for tan we can find derivative formulas for all of the other trigonometric functions. Also, recall that when we derived the General Rule for the Exponential Function we stated that we would give all derivative formulas in a general form using the Chain Rule. In this form we introduce an intermediate variable u assumed to represent some function of x. With this assumption the derivative rules for all six basic trigonometric functions are: d du sin u = cos u dx dx d cos u dx Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 8/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Derivative of sin Derivative of cos Using the Chain Rule Derivative of tan Using the Quotient Rule Derivatives the Six Trigonometric Functions Derivatives the Six Trigonometric Functions Using basic differentiation rules as in the derivation of the derivative formula for tan we can find derivative formulas for all of the other trigonometric functions. Also, recall that when we derived the General Rule for the Exponential Function we stated that we would give all derivative formulas in a general form using the Chain Rule. In this form we introduce an intermediate variable u assumed to represent some function of x. With this assumption the derivative rules for all six basic trigonometric functions are: d du sin u = cos u dx dx d du cos u = − sin u dx dx Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 8/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Derivative of sin Derivative of cos Using the Chain Rule Derivative of tan Using the Quotient Rule Derivatives the Six Trigonometric Functions Derivatives the Six Trigonometric Functions Using basic differentiation rules as in the derivation of the derivative formula for tan we can find derivative formulas for all of the other trigonometric functions. Also, recall that when we derived the General Rule for the Exponential Function we stated that we would give all derivative formulas in a general form using the Chain Rule. In this form we introduce an intermediate variable u assumed to represent some function of x. With this assumption the derivative rules for all six basic trigonometric functions are: d du sin u = cos u dx dx d tan u dx d du cos u = − sin u dx dx Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 8/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Derivative of sin Derivative of cos Using the Chain Rule Derivative of tan Using the Quotient Rule Derivatives the Six Trigonometric Functions Derivatives the Six Trigonometric Functions Using basic differentiation rules as in the derivation of the derivative formula for tan we can find derivative formulas for all of the other trigonometric functions. Also, recall that when we derived the General Rule for the Exponential Function we stated that we would give all derivative formulas in a general form using the Chain Rule. In this form we introduce an intermediate variable u assumed to represent some function of x. With this assumption the derivative rules for all six basic trigonometric functions are: d du sin u = cos u dx dx du d du tan u = sec2 u = 1 + tan2 u dx dx dx Clint Lee d du cos u = − sin u dx dx Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 8/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Derivative of sin Derivative of cos Using the Chain Rule Derivative of tan Using the Quotient Rule Derivatives the Six Trigonometric Functions Derivatives the Six Trigonometric Functions Using basic differentiation rules as in the derivation of the derivative formula for tan we can find derivative formulas for all of the other trigonometric functions. Also, recall that when we derived the General Rule for the Exponential Function we stated that we would give all derivative formulas in a general form using the Chain Rule. In this form we introduce an intermediate variable u assumed to represent some function of x. With this assumption the derivative rules for all six basic trigonometric functions are: d du sin u = cos u dx dx du d du tan u = sec2 u = 1 + tan2 u dx dx dx d sec u dx Clint Lee d du cos u = − sin u dx dx Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 8/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Derivative of sin Derivative of cos Using the Chain Rule Derivative of tan Using the Quotient Rule Derivatives the Six Trigonometric Functions Derivatives the Six Trigonometric Functions Using basic differentiation rules as in the derivation of the derivative formula for tan we can find derivative formulas for all of the other trigonometric functions. Also, recall that when we derived the General Rule for the Exponential Function we stated that we would give all derivative formulas in a general form using the Chain Rule. In this form we introduce an intermediate variable u assumed to represent some function of x. With this assumption the derivative rules for all six basic trigonometric functions are: d du sin u = cos u dx dx du d du tan u = sec2 u = 1 + tan2 u dx dx dx du d sec u = sec u tan u dx dx Clint Lee d du cos u = − sin u dx dx Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 8/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Derivative of sin Derivative of cos Using the Chain Rule Derivative of tan Using the Quotient Rule Derivatives the Six Trigonometric Functions Derivatives the Six Trigonometric Functions Using basic differentiation rules as in the derivation of the derivative formula for tan we can find derivative formulas for all of the other trigonometric functions. Also, recall that when we derived the General Rule for the Exponential Function we stated that we would give all derivative formulas in a general form using the Chain Rule. In this form we introduce an intermediate variable u assumed to represent some function of x. With this assumption the derivative rules for all six basic trigonometric functions are: d du sin u = cos u dx dx du d du tan u = sec2 u = 1 + tan2 u dx dx dx du d sec u = sec u tan u dx dx Clint Lee d du cos u = − sin u dx dx d csc u dx Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 8/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Derivative of sin Derivative of cos Using the Chain Rule Derivative of tan Using the Quotient Rule Derivatives the Six Trigonometric Functions Derivatives the Six Trigonometric Functions Using basic differentiation rules as in the derivation of the derivative formula for tan we can find derivative formulas for all of the other trigonometric functions. Also, recall that when we derived the General Rule for the Exponential Function we stated that we would give all derivative formulas in a general form using the Chain Rule. In this form we introduce an intermediate variable u assumed to represent some function of x. With this assumption the derivative rules for all six basic trigonometric functions are: d du sin u = cos u dx dx du d du tan u = sec2 u = 1 + tan2 u dx dx dx du d sec u = sec u tan u dx dx Clint Lee d du cos u = − sin u dx dx du d csc u = − csc u cot u dx dx Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 8/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Derivative of sin Derivative of cos Using the Chain Rule Derivative of tan Using the Quotient Rule Derivatives the Six Trigonometric Functions Derivatives the Six Trigonometric Functions Using basic differentiation rules as in the derivation of the derivative formula for tan we can find derivative formulas for all of the other trigonometric functions. Also, recall that when we derived the General Rule for the Exponential Function we stated that we would give all derivative formulas in a general form using the Chain Rule. In this form we introduce an intermediate variable u assumed to represent some function of x. With this assumption the derivative rules for all six basic trigonometric functions are: d dx d dx d dx d dx du dx du du tan u = sec2 u = 1 + tan2 u dx dx du sec u = sec u tan u dx sin u = cos u d du cos u = − sin u dx dx du d csc u = − csc u cot u dx dx cot u Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 8/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Derivative of sin Derivative of cos Using the Chain Rule Derivative of tan Using the Quotient Rule Derivatives the Six Trigonometric Functions Derivatives the Six Trigonometric Functions Using basic differentiation rules as in the derivation of the derivative formula for tan we can find derivative formulas for all of the other trigonometric functions. Also, recall that when we derived the General Rule for the Exponential Function we stated that we would give all derivative formulas in a general form using the Chain Rule. In this form we introduce an intermediate variable u assumed to represent some function of x. With this assumption the derivative rules for all six basic trigonometric functions are: d dx d dx d dx d dx du dx du du tan u = sec2 u = 1 + tan2 u dx dx du sec u = sec u tan u dx du 2 du cot u = − csc u = − 1 + cot2 u dx dx sin u = cos u Clint Lee d du cos u = − sin u dx dx du d csc u = − csc u cot u dx dx Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 8/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Example 46 – Differentiating with Trig Functions Example 47 – Damped Oscillations Example 46 – Differentiating with Trig Functions Find and simplify the indicated derivative(s) of each function. (a) Find f 0 (x) and f 00 (x) for f (x) = x2 cos (3x). cos t ds for s = . (b) Find dt sin t + cos t √ 2 (c) Find C0 (x) for C(x) = tan e 1+x . Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 9/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Example 46 – Differentiating with Trig Functions Example 47 – Damped Oscillations Solution: Example 46(a) Using the Product Rule followed by the Chain Rule (for cos (3x)) gives f 0 (x) Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 10/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Example 46 – Differentiating with Trig Functions Example 47 – Damped Oscillations Solution: Example 46(a) Using the Product Rule followed by the Chain Rule (for cos (3x)) gives f 0 (x) = 2x cos (3x) + x2 [−3 sin (3x)] Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 10/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Example 46 – Differentiating with Trig Functions Example 47 – Damped Oscillations Solution: Example 46(a) Using the Product Rule followed by the Chain Rule (for cos (3x)) gives f 0 (x) = 2x cos (3x) + x2 [−3 sin (3x)] = 2x cos (3x) − 3x2 sin (3x) Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 10/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Example 46 – Differentiating with Trig Functions Example 47 – Damped Oscillations Solution: Example 46(a) Using the Product Rule followed by the Chain Rule (for cos (3x)) gives f 0 (x) = 2x cos (3x) + x2 [−3 sin (3x)] = 2x cos (3x) − 3x2 sin (3x) = x [2 cos (3x) − 3x sin (3x)] Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 10/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Example 46 – Differentiating with Trig Functions Example 47 – Damped Oscillations Solution: Example 46(a) Using the Product Rule followed by the Chain Rule (for cos (3x)) gives f 0 (x) = 2x cos (3x) + x2 [−3 sin (3x)] = 2x cos (3x) − 3x2 sin (3x) = x [2 cos (3x) − 3x sin (3x)] For f 00 (x) use the expression in the second line. Again using the Product and Chain Rules gives f 00 (x) Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 10/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Example 46 – Differentiating with Trig Functions Example 47 – Damped Oscillations Solution: Example 46(a) Using the Product Rule followed by the Chain Rule (for cos (3x)) gives f 0 (x) = 2x cos (3x) + x2 [−3 sin (3x)] = 2x cos (3x) − 3x2 sin (3x) = x [2 cos (3x) − 3x sin (3x)] For f 00 (x) use the expression in the second line. Again using the Product and Chain Rules gives f 00 (x) = 2 cos (3x) − 6x sin (3x) − 6x sin (3x) − 9x2 cos (3x) Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 10/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Example 46 – Differentiating with Trig Functions Example 47 – Damped Oscillations Solution: Example 46(a) Using the Product Rule followed by the Chain Rule (for cos (3x)) gives f 0 (x) = 2x cos (3x) + x2 [−3 sin (3x)] = 2x cos (3x) − 3x2 sin (3x) = x [2 cos (3x) − 3x sin (3x)] For f 00 (x) use the expression in the second line. Again using the Product and Chain Rules gives f 00 (x) = 2 cos (3x) − 6x sin (3x) − 6x sin (3x) − 9x2 cos (3x) = 2 − 9x2 cos (3x) − 12x sin (3x) Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 10/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Example 46 – Differentiating with Trig Functions Example 47 – Damped Oscillations Solution: Example 46(b) Using the Quotient Rule gives ds dt Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 11/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Example 46 – Differentiating with Trig Functions Example 47 – Damped Oscillations Solution: Example 46(b) Using the Quotient Rule gives ds − sin t (sin t + cos t) − cos t (cos t − sin t) = dt (sin t + cos t)2 Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 11/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Example 46 – Differentiating with Trig Functions Example 47 – Damped Oscillations Solution: Example 46(b) Using the Quotient Rule gives ds − sin t (sin t + cos t) − cos t (cos t − sin t) = dt (sin t + cos t)2 = − sin2 t − sin t cos t − cos2 t + cos t sin t (sin t + cos t)2 Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 11/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Example 46 – Differentiating with Trig Functions Example 47 – Damped Oscillations Solution: Example 46(b) Using the Quotient Rule gives ds − sin t (sin t + cos t) − cos t (cos t − sin t) = dt (sin t + cos t)2 = = − sin2 t − sin t cos t − cos2 t + cos t sin t (sin t + cos t)2 − sin2 t + cos2 t (sin t + cos t)2 Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 11/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Example 46 – Differentiating with Trig Functions Example 47 – Damped Oscillations Solution: Example 46(b) Using the Quotient Rule gives ds − sin t (sin t + cos t) − cos t (cos t − sin t) = dt (sin t + cos t)2 = = − sin2 t − sin t cos t − cos2 t + cos t sin t (sin t + cos t)2 − sin2 t + cos2 t =− (sin t + cos t)2 1 (sin t + cos t)2 Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 11/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Example 46 – Differentiating with Trig Functions Example 47 – Damped Oscillations Solution: Example 46(b) Using the Quotient Rule gives ds − sin t (sin t + cos t) − cos t (cos t − sin t) = dt (sin t + cos t)2 = = − sin2 t − sin t cos t − cos2 t + cos t sin t (sin t + cos t)2 − sin2 t + cos2 t =− (sin t + cos t)2 1 (sin t + cos t)2 This example illustrates the fact that when simplifying derivatives involving trig functions, you sometimes need to use standard trigonometric identities. Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 11/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Example 46 – Differentiating with Trig Functions Example 47 – Damped Oscillations Solution: Example 46(c) This is a composite function with Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 12/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Example 46 – Differentiating with Trig Functions Example 47 – Damped Oscillations Solution: Example 46(c) This is a composite function with C(x) Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 12/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Example 46 – Differentiating with Trig Functions Example 47 – Damped Oscillations Solution: Example 46(c) This is a composite function with C(x) = f (g (h (k(x)))) Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 12/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Example 46 – Differentiating with Trig Functions Example 47 – Damped Oscillations Solution: Example 46(c) This is a composite function with C(x) = f (g (h (k(x)))) where f (x) Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 12/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Example 46 – Differentiating with Trig Functions Example 47 – Damped Oscillations Solution: Example 46(c) This is a composite function with C(x) = f (g (h (k(x)))) where f (x) = tan x, Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 12/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Example 46 – Differentiating with Trig Functions Example 47 – Damped Oscillations Solution: Example 46(c) This is a composite function with C(x) = f (g (h (k(x)))) where f (x) = tan x, g(x) Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 12/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Example 46 – Differentiating with Trig Functions Example 47 – Damped Oscillations Solution: Example 46(c) This is a composite function with C(x) = f (g (h (k(x)))) where f (x) = tan x, g(x) = ex , Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 12/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Example 46 – Differentiating with Trig Functions Example 47 – Damped Oscillations Solution: Example 46(c) This is a composite function with C(x) = f (g (h (k(x)))) where f (x) = tan x, g(x) = ex , h(x) Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 12/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Example 46 – Differentiating with Trig Functions Example 47 – Damped Oscillations Solution: Example 46(c) This is a composite function with C(x) = f (g (h (k(x)))) where f (x) = tan x, g(x) = ex , h(x) = Clint Lee √ x = x1/2 , Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 12/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Example 46 – Differentiating with Trig Functions Example 47 – Damped Oscillations Solution: Example 46(c) This is a composite function with C(x) = f (g (h (k(x)))) where f (x) = tan x, g(x) = ex , h(x) = Clint Lee √ x = x1/2 , k(x) Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 12/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Example 46 – Differentiating with Trig Functions Example 47 – Damped Oscillations Solution: Example 46(c) This is a composite function with C(x) = f (g (h (k(x)))) where f (x) = tan x, g(x) = ex , h(x) = Clint Lee √ x = x1/2 , k(x) = 1 + x2 Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 12/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Example 46 – Differentiating with Trig Functions Example 47 – Damped Oscillations Solution: Example 46(c) This is a composite function with C(x) = f (g (h (k(x)))) where f (x) = tan x, g(x) = ex , h(x) = √ x = x1/2 , k(x) = 1 + x2 Using the Chain Rule three times gives C0 (x) Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 12/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Example 46 – Differentiating with Trig Functions Example 47 – Damped Oscillations Solution: Example 46(c) This is a composite function with C(x) = f (g (h (k(x)))) where f (x) = tan x, g(x) = ex , h(x) = √ x = x1/2 , k(x) = 1 + x2 Using the Chain Rule three times gives C0 (x) = f 0 (g (h (k(x)))) g0 (h (k(x))) h0 (k(x)) k0 (x) Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 12/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Example 46 – Differentiating with Trig Functions Example 47 – Damped Oscillations Solution: Example 46(c) This is a composite function with C(x) = f (g (h (k(x)))) where f (x) = tan x, g(x) = ex , h(x) = √ x = x1/2 , k(x) = 1 + x2 Using the Chain Rule three times gives C0 (x) = f 0 (g (h (k(x)))) g0 (h (k(x))) h0 (k(x)) k0 (x) −1/2 √ 2 √ 2 1 e 1+x 1 + x2 = sec2 e 1+x (2x) 2 Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 12/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Example 46 – Differentiating with Trig Functions Example 47 – Damped Oscillations Solution: Example 46(c) This is a composite function with C(x) = f (g (h (k(x)))) where f (x) = tan x, g(x) = ex , h(x) = √ x = x1/2 , k(x) = 1 + x2 Using the Chain Rule three times gives C0 (x) = f 0 (g (h (k(x)))) g0 (h (k(x))) h0 (k(x)) k0 (x) −1/2 √ 2 √ 2 1 e 1+x 1 + x2 = sec2 e 1+x (2x) 2 √ √ 2 2 xe 1+x sec2 e 1+x √ = 1 + x2 Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 12/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Example 46 – Differentiating with Trig Functions Example 47 – Damped Oscillations Example 47 – Damped Oscillations Consider the function q(t) = e−7t sin (24t) This function describes damped simple harmonic motion. It gives the position of a mass attached to a spring relative to the equilibrium (resting) position of the spring. A frictional force acts to gradually slow the mass. (a) Find q0 (t) and q00 (t) and explain their meaning in terms of the damped oscillatory motion. (b) Note that q(0) = 0. This means that the initial position of the mass is at the equilibrium position of the spring. Find the initial velocity of the mass. Also find the velocity when the mass first returns to the equilibrium position. (c) Draw a graph of the function q(t). (d) Find the first two times when the oscillating mass turns around. Show the corresponding points on the graph of q(t). (e) Show that the function q(t) satisfies the differential equation dq d2 q + 14 + 625q = 0 dt dt2 Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 13/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Example 46 – Differentiating with Trig Functions Example 47 – Damped Oscillations Solution: Example 47(a) Using the Product and Chain Rules gives Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 14/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Example 46 – Differentiating with Trig Functions Example 47 – Damped Oscillations Solution: Example 47(a) Using the Product and Chain Rules gives q0 (t) Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 14/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Example 46 – Differentiating with Trig Functions Example 47 – Damped Oscillations Solution: Example 47(a) Using the Product and Chain Rules gives q0 (t) = −7e−7t sin (24t) + e−7t [24 cos (24t)] Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 14/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Example 46 – Differentiating with Trig Functions Example 47 – Damped Oscillations Solution: Example 47(a) Using the Product and Chain Rules gives q0 (t) = −7e−7t sin (24t) + e−7t [24 cos (24t)] = e−7t [24 cos (24t) − 7 sin (24t)] Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 14/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Example 46 – Differentiating with Trig Functions Example 47 – Damped Oscillations Solution: Example 47(a) Using the Product and Chain Rules gives q0 (t) = −7e−7t sin (24t) + e−7t [24 cos (24t)] = e−7t [24 cos (24t) − 7 sin (24t)] From our interpretation of the derivative as a rate of change, we know that this is the velocity of the mass at time t. Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 14/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Example 46 – Differentiating with Trig Functions Example 47 – Damped Oscillations Solution: Example 47(a) Using the Product and Chain Rules gives q0 (t) = −7e−7t sin (24t) + e−7t [24 cos (24t)] = e−7t [24 cos (24t) − 7 sin (24t)] From our interpretation of the derivative as a rate of change, we know that this is the velocity of the mass at time t. Taking the derivative of the expression above for q0 (t) gives Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 14/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Example 46 – Differentiating with Trig Functions Example 47 – Damped Oscillations Solution: Example 47(a) Using the Product and Chain Rules gives q0 (t) = −7e−7t sin (24t) + e−7t [24 cos (24t)] = e−7t [24 cos (24t) − 7 sin (24t)] From our interpretation of the derivative as a rate of change, we know that this is the velocity of the mass at time t. Taking the derivative of the expression above for q0 (t) gives q00 (t) Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 14/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Example 46 – Differentiating with Trig Functions Example 47 – Damped Oscillations Solution: Example 47(a) Using the Product and Chain Rules gives q0 (t) = −7e−7t sin (24t) + e−7t [24 cos (24t)] = e−7t [24 cos (24t) − 7 sin (24t)] From our interpretation of the derivative as a rate of change, we know that this is the velocity of the mass at time t. Taking the derivative of the expression above for q0 (t) gives i h q00 (t) = −7e−7t [24 cos (24t) − 7 sin (24t)] + e−7t −242 sin (24t) − 7(24) cos (24t) Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 14/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Example 46 – Differentiating with Trig Functions Example 47 – Damped Oscillations Solution: Example 47(a) Using the Product and Chain Rules gives q0 (t) = −7e−7t sin (24t) + e−7t [24 cos (24t)] = e−7t [24 cos (24t) − 7 sin (24t)] From our interpretation of the derivative as a rate of change, we know that this is the velocity of the mass at time t. Taking the derivative of the expression above for q0 (t) gives i h q00 (t) = −7e−7t [24 cos (24t) − 7 sin (24t)] + e−7t −242 sin (24t) − 7(24) cos (24t) i h = −e−7t 14(24) cos (24t) + 242 − 72 sin (24t) Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 14/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Example 46 – Differentiating with Trig Functions Example 47 – Damped Oscillations Solution: Example 47(a) Using the Product and Chain Rules gives q0 (t) = −7e−7t sin (24t) + e−7t [24 cos (24t)] = e−7t [24 cos (24t) − 7 sin (24t)] From our interpretation of the derivative as a rate of change, we know that this is the velocity of the mass at time t. Taking the derivative of the expression above for q0 (t) gives i h q00 (t) = −7e−7t [24 cos (24t) − 7 sin (24t)] + e−7t −242 sin (24t) − 7(24) cos (24t) i h = −e−7t 14(24) cos (24t) + 242 − 72 sin (24t) = −e−7t [336 cos (24t) + 527 sin (24t)] Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 14/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Example 46 – Differentiating with Trig Functions Example 47 – Damped Oscillations Solution: Example 47(a) Using the Product and Chain Rules gives q0 (t) = −7e−7t sin (24t) + e−7t [24 cos (24t)] = e−7t [24 cos (24t) − 7 sin (24t)] From our interpretation of the derivative as a rate of change, we know that this is the velocity of the mass at time t. Taking the derivative of the expression above for q0 (t) gives i h q00 (t) = −7e−7t [24 cos (24t) − 7 sin (24t)] + e−7t −242 sin (24t) − 7(24) cos (24t) i h = −e−7t 14(24) cos (24t) + 242 − 72 sin (24t) = −e−7t [336 cos (24t) + 527 sin (24t)] The rate of change of velocity is acceleration, so this gives the acceleration of the mass at time t. Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 14/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Example 46 – Differentiating with Trig Functions Example 47 – Damped Oscillations Solution: Example 47(b) Substituting t = 0 into the expression for the velocity v(t) = q0 (t) gives v(0) Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 15/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Example 46 – Differentiating with Trig Functions Example 47 – Damped Oscillations Solution: Example 47(b) Substituting t = 0 into the expression for the velocity v(t) = q0 (t) gives v(0) = e0 [24 cos (0) − 7 sin (0)] Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 15/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Example 46 – Differentiating with Trig Functions Example 47 – Damped Oscillations Solution: Example 47(b) Substituting t = 0 into the expression for the velocity v(t) = q0 (t) gives v(0) = e0 [24 cos (0) − 7 sin (0)] = 24 Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 15/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Example 46 – Differentiating with Trig Functions Example 47 – Damped Oscillations Solution: Example 47(b) Substituting t = 0 into the expression for the velocity v(t) = q0 (t) gives v(0) = e0 [24 cos (0) − 7 sin (0)] = 24 The mass returns to the equilibrium position when Clint Lee . Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 15/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Example 46 – Differentiating with Trig Functions Example 47 – Damped Oscillations Solution: Example 47(b) Substituting t = 0 into the expression for the velocity v(t) = q0 (t) gives v(0) = e0 [24 cos (0) − 7 sin (0)] = 24 The mass returns to the equilibrium position when q(t) = 0. Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 15/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Example 46 – Differentiating with Trig Functions Example 47 – Damped Oscillations Solution: Example 47(b) Substituting t = 0 into the expression for the velocity v(t) = q0 (t) gives v(0) = e0 [24 cos (0) − 7 sin (0)] = 24 The mass returns to the equilibrium position when q(t) = 0. The first time after t = 0 when this happens is when Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 15/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Example 46 – Differentiating with Trig Functions Example 47 – Damped Oscillations Solution: Example 47(b) Substituting t = 0 into the expression for the velocity v(t) = q0 (t) gives v(0) = e0 [24 cos (0) − 7 sin (0)] = 24 The mass returns to the equilibrium position when q(t) = 0. The first time after t = 0 when this happens is when 24t = π ⇒ t Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 15/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Example 46 – Differentiating with Trig Functions Example 47 – Damped Oscillations Solution: Example 47(b) Substituting t = 0 into the expression for the velocity v(t) = q0 (t) gives v(0) = e0 [24 cos (0) − 7 sin (0)] = 24 The mass returns to the equilibrium position when q(t) = 0. The first time after t = 0 when this happens is when 24t = π ⇒ t = Clint Lee π 24 Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 15/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Example 46 – Differentiating with Trig Functions Example 47 – Damped Oscillations Solution: Example 47(b) Substituting t = 0 into the expression for the velocity v(t) = q0 (t) gives v(0) = e0 [24 cos (0) − 7 sin (0)] = 24 The mass returns to the equilibrium position when q(t) = 0. The first time after t = 0 when this happens is when 24t = π ⇒ t = π 24 The velocity at this time is π v 24 Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 15/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Example 46 – Differentiating with Trig Functions Example 47 – Damped Oscillations Solution: Example 47(b) Substituting t = 0 into the expression for the velocity v(t) = q0 (t) gives v(0) = e0 [24 cos (0) − 7 sin (0)] = 24 The mass returns to the equilibrium position when q(t) = 0. The first time after t = 0 when this happens is when 24t = π ⇒ t = π 24 The velocity at this time is π v = e−7π/24 [24 cos (π ) − 7 sin (π )] 24 Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 15/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Example 46 – Differentiating with Trig Functions Example 47 – Damped Oscillations Solution: Example 47(b) Substituting t = 0 into the expression for the velocity v(t) = q0 (t) gives v(0) = e0 [24 cos (0) − 7 sin (0)] = 24 The mass returns to the equilibrium position when q(t) = 0. The first time after t = 0 when this happens is when 24t = π ⇒ t = π 24 The velocity at this time is π v = e−7π/24 [24 cos (π ) − 7 sin (π )] = −24e−7π/24 24 Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 15/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Example 46 – Differentiating with Trig Functions Example 47 – Damped Oscillations Solution: Example 47(b) Substituting t = 0 into the expression for the velocity v(t) = q0 (t) gives v(0) = e0 [24 cos (0) − 7 sin (0)] = 24 The mass returns to the equilibrium position when q(t) = 0. The first time after t = 0 when this happens is when 24t = π ⇒ t = π 24 The velocity at this time is π v = e−7π/24 [24 cos (π ) − 7 sin (π )] = −24e−7π/24 = −9.6 24 Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 15/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Example 46 – Differentiating with Trig Functions Example 47 – Damped Oscillations Solution: Example 47(b) Substituting t = 0 into the expression for the velocity v(t) = q0 (t) gives v(0) = e0 [24 cos (0) − 7 sin (0)] = 24 The mass returns to the equilibrium position when q(t) = 0. The first time after t = 0 when this happens is when 24t = π ⇒ t = π 24 The velocity at this time is π v = e−7π/24 [24 cos (π ) − 7 sin (π )] = −24e−7π/24 = −9.6 24 This velocity is less than the initial velocity and in the opposite direction. Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 15/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Example 46 – Differentiating with Trig Functions Example 47 – Damped Oscillations Solution: Example 47(c) The graph of the function q(t) looks like this. The amplitude of the motion decreases following an envelope given by the decaying exponential function e−7t , as shown. 0.1 Clint Lee 0.2 0.3 0.4 Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 16/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Example 46 – Differentiating with Trig Functions Example 47 – Damped Oscillations Solution: Example 47(c) The graph of the function q(t) looks like this. The amplitude of the motion decreases following an envelope given by the decaying exponential function e−7t , as shown. As we saw in the last part, not only does the amplitude decrease, but so does the velocity. 0.1 Clint Lee 0.2 0.3 0.4 Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 16/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Example 46 – Differentiating with Trig Functions Example 47 – Damped Oscillations Solution: Example 47(c) The graph of the function q(t) looks like this. The amplitude of the motion decreases following an envelope given by the decaying exponential function e−7t , as shown. As we saw in the last part, not only does the amplitude decrease, but so does the velocity. Further, note that where the graph is concave down, q00 (t) < 0, the mass is decelerating. The velocity is getting less positive or more negative. Clint Lee 0.1 0.2 0.3 0.4 Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 16/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Example 46 – Differentiating with Trig Functions Example 47 – Damped Oscillations Solution: Example 47(c) The graph of the function q(t) looks like this. The amplitude of the motion decreases following an envelope given by the decaying exponential function e−7t , as shown. As we saw in the last part, not only does the amplitude decrease, but so does the velocity. Further, note that where the graph is concave down, q00 (t) < 0, the mass is decelerating. The velocity is getting less positive or more negative. And, where the graph is concave up, q00 (t) > 0, the mass is accelerating. The velocity is getting more positive or less negative. Clint Lee 0.1 0.2 0.3 0.4 Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 16/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Example 46 – Differentiating with Trig Functions Example 47 – Damped Oscillations Solution: Example 47(d) The mass turns around when v(t) = q0 (t) = 0. This happens when Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 17/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Example 46 – Differentiating with Trig Functions Example 47 – Damped Oscillations Solution: Example 47(d) The mass turns around when v(t) = q0 (t) = 0. This happens when 24 cos (24t) − 7 sin (24t) = 0 0.1 Clint Lee 0.2 0.3 0.4 Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 17/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Example 46 – Differentiating with Trig Functions Example 47 – Damped Oscillations Solution: Example 47(d) The mass turns around when v(t) = q0 (t) = 0. This happens when 24 cos (24t) − 7 sin (24t) = 0 ⇒ tan (24t) = 24 7 0.1 Clint Lee 0.2 0.3 0.4 Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 17/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Example 46 – Differentiating with Trig Functions Example 47 – Damped Oscillations Solution: Example 47(d) The mass turns around when v(t) = q0 (t) = 0. This happens when 24 cos (24t) − 7 sin (24t) = 0 ⇒ tan (24t) = ⇒24t = arctan 24 7 24 7 + kπ Clint Lee 0.1 0.2 0.3 0.4 Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 17/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Example 46 – Differentiating with Trig Functions Example 47 – Damped Oscillations Solution: Example 47(d) The mass turns around when v(t) = q0 (t) = 0. This happens when 24 cos (24t) − 7 sin (24t) = 0 ⇒ tan (24t) = ⇒24t = arctan 24 7 24 7 + kπ 0.1 0.2 0.3 0.4 where k is any integer. Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 17/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Example 46 – Differentiating with Trig Functions Example 47 – Damped Oscillations Solution: Example 47(d) The mass turns around when v(t) = q0 (t) = 0. This happens when 24 cos (24t) − 7 sin (24t) = 0 ⇒ tan (24t) = ⇒24t = arctan 24 7 24 7 + kπ 0.1 0.2 0.3 0.4 where k is any integer. The first two positive t values are t1 = 0.0536 and t2 = 0.1845. Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 17/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Example 46 – Differentiating with Trig Functions Example 47 – Damped Oscillations Solution: Example 47(d) The mass turns around when v(t) = q0 (t) = 0. This happens when 24 cos (24t) − 7 sin (24t) = 0 ⇒ tan (24t) = ⇒24t = arctan 24 7 24 7 + kπ 0.1 0.2 0.3 0.4 where k is any integer. The first two positive t values are t1 = 0.0536 and t2 = 0.1845. At these values the tangent line to the graph of q(t) is horizontal, as shown. Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 17/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Example 46 – Differentiating with Trig Functions Example 47 – Damped Oscillations Solution: Example 47(d) The mass turns around when v(t) = q0 (t) = 0. This happens when 24 cos (24t) − 7 sin (24t) = 0 ⇒ tan (24t) = ⇒24t = arctan 24 7 24 7 + kπ 0.1 0.2 0.3 0.4 where k is any integer. The first two positive t values are t1 = 0.0536 and t2 = 0.1845. At these values the tangent line to the graph of q(t) is horizontal, as shown. Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 17/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Example 46 – Differentiating with Trig Functions Example 47 – Damped Oscillations Solution: Example 47(e) From part (a) we have d2 q dq + 14 + 625q dt dt2 Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 18/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Example 46 – Differentiating with Trig Functions Example 47 – Damped Oscillations Solution: Example 47(e) From part (a) we have d2 q dq + 14 + 625q dt dt2 = −e−7t [336 cos (24t) + 527 sin (24t)] Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 18/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Example 46 – Differentiating with Trig Functions Example 47 – Damped Oscillations Solution: Example 47(e) From part (a) we have d2 q dq + 14 + 625q dt dt2 = −e−7t [336 cos (24t) + 527 sin (24t)] + 14e−7t [24 cos (24t) − 7 sin (24t)] + 625e−7t sin (24t) Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 18/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Example 46 – Differentiating with Trig Functions Example 47 – Damped Oscillations Solution: Example 47(e) From part (a) we have d2 q dq + 14 + 625q dt dt2 = −e−7t [336 cos (24t) + 527 sin (24t)] + 14e−7t [24 cos (24t) − 7 sin (24t)] + 625e−7t sin (24t) = e−7t [(−336 + 14 × 24) cos (24t) + (−527 − 14 × 7 + 625) sin (24t)] Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 18/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions Example 46 – Differentiating with Trig Functions Example 47 – Damped Oscillations Solution: Example 47(e) From part (a) we have d2 q dq + 14 + 625q dt dt2 = −e−7t [336 cos (24t) + 527 sin (24t)] + 14e−7t [24 cos (24t) − 7 sin (24t)] + 625e−7t sin (24t) = e−7t [(−336 + 14 × 24) cos (24t) + (−527 − 14 × 7 + 625) sin (24t)] =0 Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 18/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions The arctan Function The arcsin Function Example 48 – Differentiating with Inverse Trig Functions The arctan Function Recall that the graph of the tangent function looks like this. Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 19/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions The arctan Function The arcsin Function Example 48 – Differentiating with Inverse Trig Functions The arctan Function Recall that the graph of the tangent function looks like this. From this graph we realize that the tangent function is not one-to-one − π2 Clint Lee π 2 Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 19/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions The arctan Function The arcsin Function Example 48 – Differentiating with Inverse Trig Functions The arctan Function Recall that the graph of the tangent function looks like this. From this graph we realize that the tangent function is not one-to-one, and so does not have an inverse function. − π2 Clint Lee π 2 Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 19/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions The arctan Function The arcsin Function Example 48 – Differentiating with Inverse Trig Functions The arctan Function Recall that the graph of the tangent function looks like this. From this graph we realize that the tangent function is not one-to-one, and so does not have an inverse function. However, restricting the domain of the tangent function, Clint Lee − π2 π 2 Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 19/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions The arctan Function The arcsin Function Example 48 – Differentiating with Inverse Trig Functions The arctan Function Recall that the graph of the tangent function looks like this. From this graph we realize that the tangent function is not one-to-one, and so does not have an inverse function. However, restricting the domain of the tangent function, as shown in the graph, to the interval − − π2 π 2 π π <x< 2 2 Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 19/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions The arctan Function The arcsin Function Example 48 – Differentiating with Inverse Trig Functions The arctan Function Recall that the graph of the tangent function looks like this. From this graph we realize that the tangent function is not one-to-one, and so does not have an inverse function. However, restricting the domain of the tangent function, as shown in the graph, to the interval − − π2 π 2 π π <x< 2 2 gives a one-to-one function. Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 19/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions The arctan Function The arcsin Function Example 48 – Differentiating with Inverse Trig Functions The arctan Function Recall that the graph of the tangent function looks like this. From this graph we realize that the tangent function is not one-to-one, and so does not have an inverse function. However, restricting the domain of the tangent function, as shown in the graph, to the interval − − π2 π 2 π π <x< 2 2 gives a one-to-one function. So the inverse of the tangent function is defined as arctan x = tan−1 x = the angle between − π2 and Clint Lee π 2 whose tangent is x Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 19/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions The arctan Function The arcsin Function Example 48 – Differentiating with Inverse Trig Functions The Derivative of the arctan Function With this definition of the inverse tangent function, we see that its domain is Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 20/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions The arctan Function The arcsin Function Example 48 – Differentiating with Inverse Trig Functions The Derivative of the arctan Function With this definition of the inverse tangent function, we see that its domain is all real numbers and its range is Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 20/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions The arctan Function The arcsin Function Example 48 – Differentiating with Inverse Trig Functions The Derivative of the arctan Function With this definition of the inverse tangent function, we see that its domain is all real numbers and its range is − π2 , π2 . Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 20/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions The arctan Function The arcsin Function Example 48 – Differentiating with Inverse Trig Functions The Derivative of the arctan Function With this definition of the inverse tangent function, we see that its domain is all real numbers and its range is − π2 , π2 . Further, we have tan (arctan x) Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 20/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions The arctan Function The arcsin Function Example 48 – Differentiating with Inverse Trig Functions The Derivative of the arctan Function With this definition of the inverse tangent function, we see that its domain is all real numbers and its range is − π2 , π2 . Further, we have tan (arctan x) = x and Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 20/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions The arctan Function The arcsin Function Example 48 – Differentiating with Inverse Trig Functions The Derivative of the arctan Function With this definition of the inverse tangent function, we see that its domain is all real numbers and its range is − π2 , π2 . Further, we have tan (arctan x) = x and arctan (tan x) Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 20/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions The arctan Function The arcsin Function Example 48 – Differentiating with Inverse Trig Functions The Derivative of the arctan Function With this definition of the inverse tangent function, we see that its domain is all real numbers and its range is − π2 , π2 . Further, we have tan (arctan x) = x and arctan (tan x) = x for − π2 < x < Clint Lee π 2 Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 20/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions The arctan Function The arcsin Function Example 48 – Differentiating with Inverse Trig Functions The Derivative of the arctan Function With this definition of the inverse tangent function, we see that its domain is all real numbers and its range is − π2 , π2 . Further, we have tan (arctan x) = x and arctan (tan x) = x for − π2 < x < π 2 Taking the derivative of the first of the formulas above, as we did to find the derivative of the ln function, gives d tan (arctan x) dx Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 20/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions The arctan Function The arcsin Function Example 48 – Differentiating with Inverse Trig Functions The Derivative of the arctan Function With this definition of the inverse tangent function, we see that its domain is all real numbers and its range is − π2 , π2 . Further, we have tan (arctan x) = x and arctan (tan x) = x for − π2 < x < π 2 Taking the derivative of the first of the formulas above, as we did to find the derivative of the ln function, gives d d tan (arctan x) = 1 + tan2 (arctan x) arctan x dx dx Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 20/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions The arctan Function The arcsin Function Example 48 – Differentiating with Inverse Trig Functions The Derivative of the arctan Function With this definition of the inverse tangent function, we see that its domain is all real numbers and its range is − π2 , π2 . Further, we have tan (arctan x) = x and arctan (tan x) = x for − π2 < x < π 2 Taking the derivative of the first of the formulas above, as we did to find the derivative of the ln function, gives d d tan (arctan x) = 1 + tan2 (arctan x) arctan x dx dx d = 1 + x2 arctan x dx Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 20/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions The arctan Function The arcsin Function Example 48 – Differentiating with Inverse Trig Functions The Derivative of the arctan Function With this definition of the inverse tangent function, we see that its domain is all real numbers and its range is − π2 , π2 . Further, we have tan (arctan x) = x and arctan (tan x) = x for − π2 < x < π 2 Taking the derivative of the first of the formulas above, as we did to find the derivative of the ln function, gives d d tan (arctan x) = 1 + tan2 (arctan x) arctan x dx dx d d = 1 + x2 arctan x = x=1 dx dx Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 20/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions The arctan Function The arcsin Function Example 48 – Differentiating with Inverse Trig Functions The Derivative of the arctan Function With this definition of the inverse tangent function, we see that its domain is all real numbers and its range is − π2 , π2 . Further, we have tan (arctan x) = x and arctan (tan x) = x for − π2 < x < π 2 Taking the derivative of the first of the formulas above, as we did to find the derivative of the ln function, gives d d tan (arctan x) = 1 + tan2 (arctan x) arctan x dx dx d d = 1 + x2 arctan x = x=1 dx dx Solving for d arctan x gives dx d arctan x dx Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 20/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions The arctan Function The arcsin Function Example 48 – Differentiating with Inverse Trig Functions The Derivative of the arctan Function With this definition of the inverse tangent function, we see that its domain is all real numbers and its range is − π2 , π2 . Further, we have tan (arctan x) = x and arctan (tan x) = x for − π2 < x < π 2 Taking the derivative of the first of the formulas above, as we did to find the derivative of the ln function, gives d d tan (arctan x) = 1 + tan2 (arctan x) arctan x dx dx d d = 1 + x2 arctan x = x=1 dx dx Solving for d arctan x gives dx d 1 arctan x = dx 1 + x2 Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 20/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions The arctan Function The arcsin Function Example 48 – Differentiating with Inverse Trig Functions The arcsin Function Recall that the graph of the sine function looks like this. Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 21/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions The arctan Function The arcsin Function Example 48 – Differentiating with Inverse Trig Functions The arcsin Function Recall that the graph of the sine function looks like this. From this graph we realize that the sine function is not one-to-one 1 − π2 π 2 −1 Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 21/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions The arctan Function The arcsin Function Example 48 – Differentiating with Inverse Trig Functions The arcsin Function Recall that the graph of the sine function looks like this. From this graph we realize that the sine function is not one-to-one, and so does not have an inverse function. 1 − π2 π 2 −1 Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 21/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions The arctan Function The arcsin Function Example 48 – Differentiating with Inverse Trig Functions The arcsin Function Recall that the graph of the sine function looks like this. From this graph we realize that the sine function is not one-to-one, and so does not have an inverse function. However, restricting the domain of the sine function, 1 − π2 π 2 −1 Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 21/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions The arctan Function The arcsin Function Example 48 – Differentiating with Inverse Trig Functions The arcsin Function Recall that the graph of the sine function looks like this. From this graph we realize that the sine function is not one-to-one, and so does not have an inverse function. However, restricting the domain of the sine function, as shown in the graph, to the interval − 1 − π2 π 2 −1 π π ≤x≤ 2 2 Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 21/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions The arctan Function The arcsin Function Example 48 – Differentiating with Inverse Trig Functions The arcsin Function Recall that the graph of the sine function looks like this. From this graph we realize that the sine function is not one-to-one, and so does not have an inverse function. However, restricting the domain of the sine function, as shown in the graph, to the interval − 1 − π2 π 2 −1 π π ≤x≤ 2 2 gives a one-to-one function. Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 21/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions The arctan Function The arcsin Function Example 48 – Differentiating with Inverse Trig Functions The arcsin Function Recall that the graph of the sine function looks like this. From this graph we realize that the sine function is not one-to-one, and so does not have an inverse function. However, restricting the domain of the sine function, as shown in the graph, to the interval − 1 − π2 π 2 −1 π π ≤x≤ 2 2 gives a one-to-one function. So the inverse of the sine function is defined as arcsin x = sin−1 x = the angle between − π2 and Clint Lee π 2 whose sin is x Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 21/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions The arctan Function The arcsin Function Example 48 – Differentiating with Inverse Trig Functions The arcsin Function Recall that the graph of the sine function looks like this. From this graph we realize that the sine function is not one-to-one, and so does not have an inverse function. However, restricting the domain of the sine function, as shown in the graph, to the interval − 1 − π2 π 2 −1 π π ≤x≤ 2 2 gives a one-to-one function. So the inverse of the sine function is defined as arcsin x = sin−1 x = the angle between − π2 and π 2 whose sin is x The domain of inverse sine function, so defined, is Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 21/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions The arctan Function The arcsin Function Example 48 – Differentiating with Inverse Trig Functions The arcsin Function Recall that the graph of the sine function looks like this. From this graph we realize that the sine function is not one-to-one, and so does not have an inverse function. However, restricting the domain of the sine function, as shown in the graph, to the interval − 1 − π2 π 2 −1 π π ≤x≤ 2 2 gives a one-to-one function. So the inverse of the sine function is defined as arcsin x = sin−1 x = the angle between − π2 and π 2 whose sin is x The domain of inverse sine function, so defined, is [−1, 1] and its range is Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 21/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions The arctan Function The arcsin Function Example 48 – Differentiating with Inverse Trig Functions The arcsin Function Recall that the graph of the sine function looks like this. From this graph we realize that the sine function is not one-to-one, and so does not have an inverse function. However, restricting the domain of the sine function, as shown in the graph, to the interval − 1 − π2 π 2 −1 π π ≤x≤ 2 2 gives a one-to-one function. So the inverse of the sine function is defined as arcsin x = sin−1 x = the angle between − π2 and π 2 whose sin is x domain of inverse sine function, so defined, is [−1, 1] and its range is The − π2 , π2 . Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 21/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions The arctan Function The arcsin Function Example 48 – Differentiating with Inverse Trig Functions The Derivative of the arcsin Function As with the arctan function, we have sin (arcsin x) Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 22/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions The arctan Function The arcsin Function Example 48 – Differentiating with Inverse Trig Functions The Derivative of the arcsin Function As with the arctan function, we have sin (arcsin x) = x and Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 22/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions The arctan Function The arcsin Function Example 48 – Differentiating with Inverse Trig Functions The Derivative of the arcsin Function As with the arctan function, we have sin (arcsin x) = x and arcsin (sin x) Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 22/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions The arctan Function The arcsin Function Example 48 – Differentiating with Inverse Trig Functions The Derivative of the arcsin Function As with the arctan function, we have sin (arcsin x) = x and arcsin (sin x) = x Clint Lee for − π2 ≤ x ≤ π 2 Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 22/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions The arctan Function The arcsin Function Example 48 – Differentiating with Inverse Trig Functions The Derivative of the arcsin Function As with the arctan function, we have sin (arcsin x) = x and arcsin (sin x) = x for − π2 ≤ x ≤ π 2 Taking the derivative of the first of the formulas above, as we just did for the arctan, gives d sin (arcsin x) dx Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 22/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions The arctan Function The arcsin Function Example 48 – Differentiating with Inverse Trig Functions The Derivative of the arcsin Function As with the arctan function, we have sin (arcsin x) = x and arcsin (sin x) = x for − π2 ≤ x ≤ π 2 Taking the derivative of the first of the formulas above, as we just did for the arctan, gives d d sin (arcsin x) = cos (arcsin x) arcsin x dx dx Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 22/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions The arctan Function The arcsin Function Example 48 – Differentiating with Inverse Trig Functions The Derivative of the arcsin Function As with the arctan function, we have sin (arcsin x) = x and arcsin (sin x) = x for − π2 ≤ x ≤ π 2 Taking the derivative of the first of the formulas above, as we just did for the arctan, gives d d d sin (arcsin x) = cos (arcsin x) arcsin x = x=1 dx dx dx Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 22/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions The arctan Function The arcsin Function Example 48 – Differentiating with Inverse Trig Functions The Derivative of the arcsin Function As with the arctan function, we have sin (arcsin x) = x and arcsin (sin x) = x for − π2 ≤ x ≤ π 2 Taking the derivative of the first of the formulas above, as we just did for the arctan, gives d d d sin (arcsin x) = cos (arcsin x) arcsin x = x=1 dx dx dx Now for − π2 ≤ x ≤ π 2 we have cos x ≥ 0 Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 22/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions The arctan Function The arcsin Function Example 48 – Differentiating with Inverse Trig Functions The Derivative of the arcsin Function As with the arctan function, we have sin (arcsin x) = x and arcsin (sin x) = x for − π2 ≤ x ≤ π 2 Taking the derivative of the first of the formulas above, as we just did for the arctan, gives d d d sin (arcsin x) = cos (arcsin x) arcsin x = x=1 dx dx dx Now for − π2 ≤ x ≤ identity cos2 x + sin we have cos x ≥ 0 , and using the basic Pythagorean p x = 1, gives cos x = 1 − sin2 x. π 2 2 Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 22/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions The arctan Function The arcsin Function Example 48 – Differentiating with Inverse Trig Functions The Derivative of the arcsin Function As with the arctan function, we have sin (arcsin x) = x and arcsin (sin x) = x for − π2 ≤ x ≤ π 2 Taking the derivative of the first of the formulas above, as we just did for the arctan, gives d d d sin (arcsin x) = cos (arcsin x) arcsin x = x=1 dx dx dx Now for − π2 ≤ x ≤ identity cos2 x + sin d sin (arcsin x) dx we have cos x ≥ 0 , and using the basic Pythagorean p x = 1, gives cos x = 1 − sin2 x. So that π 2 2 Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 22/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions The arctan Function The arcsin Function Example 48 – Differentiating with Inverse Trig Functions The Derivative of the arcsin Function As with the arctan function, we have sin (arcsin x) = x and arcsin (sin x) = x for − π2 ≤ x ≤ π 2 Taking the derivative of the first of the formulas above, as we just did for the arctan, gives d d d sin (arcsin x) = cos (arcsin x) arcsin x = x=1 dx dx dx we have cos x ≥ 0 , and using the basic Pythagorean p identity x = 1, gives cos x = 1 − sin2 x. So that q d d sin (arcsin x) = 1 − sin2 (arcsin x) arcsin x dx dx Now for − π2 ≤ x ≤ cos2 x + sin π 2 2 Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 22/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions The arctan Function The arcsin Function Example 48 – Differentiating with Inverse Trig Functions The Derivative of the arcsin Function As with the arctan function, we have sin (arcsin x) = x and arcsin (sin x) = x for − π2 ≤ x ≤ π 2 Taking the derivative of the first of the formulas above, as we just did for the arctan, gives d d d sin (arcsin x) = cos (arcsin x) arcsin x = x=1 dx dx dx we have cos x ≥ 0 , and using the basic Pythagorean p identity x = 1, gives cos x = 1 − sin2 x. So that q p d d d sin (arcsin x) = 1 − sin2 (arcsin x) arcsin x = 1 − x2 arcsin x dx dx dx Now for − π2 ≤ x ≤ cos2 x + sin π 2 2 Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 22/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions The arctan Function The arcsin Function Example 48 – Differentiating with Inverse Trig Functions The Derivative of the arcsin Function As with the arctan function, we have sin (arcsin x) = x and arcsin (sin x) = x for − π2 ≤ x ≤ π 2 Taking the derivative of the first of the formulas above, as we just did for the arctan, gives d d d sin (arcsin x) = cos (arcsin x) arcsin x = x=1 dx dx dx we have cos x ≥ 0 , and using the basic Pythagorean p identity x = 1, gives cos x = 1 − sin2 x. So that q p d d d sin (arcsin x) = 1 − sin2 (arcsin x) arcsin x = 1 − x2 arcsin x dx dx dx Now for − π2 ≤ x ≤ cos2 x + sin Solving for π 2 2 d arcsin x gives dx d arcsin x dx Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 22/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions The arctan Function The arcsin Function Example 48 – Differentiating with Inverse Trig Functions The Derivative of the arcsin Function As with the arctan function, we have sin (arcsin x) = x and arcsin (sin x) = x for − π2 ≤ x ≤ π 2 Taking the derivative of the first of the formulas above, as we just did for the arctan, gives d d d sin (arcsin x) = cos (arcsin x) arcsin x = x=1 dx dx dx we have cos x ≥ 0 , and using the basic Pythagorean p identity x = 1, gives cos x = 1 − sin2 x. So that q p d d d sin (arcsin x) = 1 − sin2 (arcsin x) arcsin x = 1 − x2 arcsin x dx dx dx Now for − π2 ≤ x ≤ cos2 x + sin Solving for π 2 2 d arcsin x gives dx 1 d arcsin x = √ dx 1 − x2 Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 22/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions The arctan Function The arcsin Function Example 48 – Differentiating with Inverse Trig Functions Example 48 – Differentiating with Inverse Trig Functions Find and simplify the indicated derivative(s) of each function. (a) Find f 0 (x) and f 00 (x) for f (x) = 1 + x2 arctan x. √ dy for y = arcsin (b) Find 1 − x2 . dx Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 23/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions The arctan Function The arcsin Function Example 48 – Differentiating with Inverse Trig Functions Solution: Example 48(a) Using the Product Rule gives f 0 (x) Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 24/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions The arctan Function The arcsin Function Example 48 – Differentiating with Inverse Trig Functions Solution: Example 48(a) Using the Product Rule gives 0 f (x) = 2x arctan x + 1 + x Clint Lee 2 1 1 + x2 Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 24/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions The arctan Function The arcsin Function Example 48 – Differentiating with Inverse Trig Functions Solution: Example 48(a) Using the Product Rule gives 0 f (x) = 2x arctan x + 1 + x Clint Lee 2 1 1 + x2 = 2x arctan x + 1 Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 24/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions The arctan Function The arcsin Function Example 48 – Differentiating with Inverse Trig Functions Solution: Example 48(a) Using the Product Rule gives 0 f (x) = 2x arctan x + 1 + x 2 1 1 + x2 = 2x arctan x + 1 Taking the derivative of the expression above, using the Product Rule again, gives f 00 (x) Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 24/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions The arctan Function The arcsin Function Example 48 – Differentiating with Inverse Trig Functions Solution: Example 48(a) Using the Product Rule gives 0 f (x) = 2x arctan x + 1 + x 2 1 1 + x2 = 2x arctan x + 1 Taking the derivative of the expression above, using the Product Rule again, gives 1 f 00 (x) = 2 arctan x + 2x 1 + x2 Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 24/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions The arctan Function The arcsin Function Example 48 – Differentiating with Inverse Trig Functions Solution: Example 48(a) Using the Product Rule gives 0 f (x) = 2x arctan x + 1 + x 2 1 1 + x2 = 2x arctan x + 1 Taking the derivative of the expression above, using the Product Rule again, gives 1 2x f 00 (x) = 2 arctan x + 2x = 2 arctan x + 1 + x2 1 + x2 Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 24/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions The arctan Function The arcsin Function Example 48 – Differentiating with Inverse Trig Functions Solution: Example 48(b) Using the Chain Rule gives dy dx Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 25/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions The arctan Function The arcsin Function Example 48 – Differentiating with Inverse Trig Functions Solution: Example 48(b) Using the Chain Rule gives dy = r dx 1− 1 √ 1 − x2 2 −1/2 1 1 − x2 (−2x) 2 Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 25/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions The arctan Function The arcsin Function Example 48 – Differentiating with Inverse Trig Functions Solution: Example 48(b) Using the Chain Rule gives dy = r dx = p 1− 1 √ 1 2 1 − x2 ! 1 − ( 1 − x2 ) √ −1/2 1 1 − x2 (−2x) 2 −x 1 − x2 Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 25/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions The arctan Function The arcsin Function Example 48 – Differentiating with Inverse Trig Functions Solution: Example 48(b) Using the Chain Rule gives dy = r dx = p 1− 1 √ 1 2 1 − x2 ! 1 − ( 1 − x2 ) √ −1/2 1 1 − x2 (−2x) 2 −x 1 − x2 Clint Lee = 1 √ x2 √ −x 1 − x2 Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 25/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions The arctan Function The arcsin Function Example 48 – Differentiating with Inverse Trig Functions Solution: Example 48(b) Using the Chain Rule gives dy = r dx = p 1− 1 √ 1 2 1 − x2 ! 1 − ( 1 − x2 ) √ −1/2 1 1 − x2 (−2x) 2 −x 1 − x2 Clint Lee = 1 √ x2 √ −x 1 − x2 =− √ x | x | 1 − x2 Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 25/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions The arctan Function The arcsin Function Example 48 – Differentiating with Inverse Trig Functions Solution: Example 48(b) Using the Chain Rule gives dy = r dx = p 1− 1 √ 1 2 1 − x2 ! 1 − ( 1 − x2 ) Thus, if 0 ≤ x < 1, then √ −1/2 1 1 − x2 (−2x) 2 −x 1 − x2 = 1 √ x2 √ −x 1 − x2 =− √ x | x | 1 − x2 p 1 d arcsin 1 − x2 = − √ dx 1 − x2 Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 25/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions The arctan Function The arcsin Function Example 48 – Differentiating with Inverse Trig Functions Solution: Example 48(b) Using the Chain Rule gives dy = r dx = p 1− 1 √ 1 2 1 − x2 ! 1 − ( 1 − x2 ) Thus, if 0 ≤ x < 1, then √ −1/2 1 1 − x2 (−2x) 2 −x 1 − x2 = 1 √ x2 √ −x 1 − x2 =− √ x | x | 1 − x2 p 1 d arcsin 1 − x2 = − √ dx 1 − x2 But you can show that d 1 arccos x = − √ dx 1 − x2 Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 25/25 Derivatives of the Basic Trigonometric Functions Applying the Trig Function Derivative Rules Derivatives of the Inverse Trigonometric Functions The arctan Function The arcsin Function Example 48 – Differentiating with Inverse Trig Functions Solution: Example 48(b) Using the Chain Rule gives dy = r dx = p 1− 1 √ 1 2 1 − x2 ! 1 − ( 1 − x2 ) Thus, if 0 ≤ x < 1, then √ −1/2 1 1 − x2 (−2x) 2 −x 1 − x2 = 1 √ x2 √ −x 1 − x2 =− √ x | x | 1 − x2 p 1 d arcsin 1 − x2 = − √ dx 1 − x2 But you can show that d 1 arccos x = − √ dx 1 − x2 √ 1 − x2 ? So is it true that arccos x = arcsin Clint Lee Math 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions 25/25
