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3.2 Differentiation Formulas
Now that we know how to find derivatives the hard way, we can learn how to differentiate the easy way
using the following rules.
(1) If f (x) = c (a constant function), then f ′ (x) = 0
(2) Power Rule: If f (x) = xn , then f ′ (x) = nxn−1
(3) If f (x) = cg(x), then f ′ (x) = cg ′ (x)
(4) If f (x) = g(x) ± h(x), then f ′ (x) = g ′ (x) ± h′ (x)
(5) Product Rule: If f (x) = g(x)h(x), then f ′ (x) = g ′ (x)h(x) + g(x)h′ (x)
(6) Quotient Rule: If f (x) =
g(x)
h(x)g ′ (x) − g(x)h′ (x)
, then f ′ (x) =
h(x)
[h(x)]2
Note: If a problem says to use the definition of the derivative, you may NOT use these rules.
Find the derivatives of the following functions.
• f (x) = 3x5 − 6x4 + 7x + 5 + e2 + 3π
• g(t) =
√
2
1
t+ √ +
t
t
• f (x) =
√
4
3
4
2/3
−3/4
9
√
−
+
5x
x
+
x
5
x x3
1
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• g(s) =
s3 + s
s2 − 3
√
t3 − 3t2 + t t
√
• g(t) =
t
√
1
Example: An object moving in a straight line has position function f (t) = 2 − 2t3 − 8t t where position
5t
is measured in meters and time is measured in seconds. Find the velocity at time t = 1.
Example: Consider the function f (x) = x3 − 3x2 − 9x + 1. For what values of x does f (x) have a horizontal
tangent?
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Example: Find an equation of the tangent line to the graph of g(x) =
Example: Suppose h(x) = f (x)g(x) and u(x) =
f (x)
, where the graphs of f and g are given below. Find
g(x)
the following.
(a) h′ (3)
(b) u′ (1)
(c) Find a formula for k′ (x) if k(x) =
3x
at x = 2.
x2 − 1
(3x + 1)2 f (x)
2g(x)
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Example: For what values of a and b is the line y = −3x + b tangent to the graph of f (x) = ax3 when
x = −2?
Example: Find the equations of both tangent lines to the curve y = x2 + 3x that also pass through the point
(−1, −6).
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Example: Where is the function f (x) = x1/3 not differentiable?
Example: Where is the function f (x) = |3x + 7| not differentiable? What is f ′ (x)?
Example: Where is the function below not differentiable? Find f ′ (x).
f (x) =


−2x − 1


 x2

3x − 2


 2
x − 5x
if
if
if
if
x < −1
−1≤x≤2
2<x<4
x≥4
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For what values of b and c is the following function differentiable everywhere?
f (x) =
(
x2
bx3 + c
if x ≤ 2
if x > 2
3.3 Rates of Change in the Natural and Social Sciences
An object is thrown vertically upward with a velocity of 40 ft/s. Its height after t seconds is given by
s(t) = 40t − 16t2 + 24.
• What is the maximum height achieved by the object?
• What is the object’s velocity when it is at a height of 40 ft on its way down?
• With what velocity does the object hit the ground?
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A particle moves according to the position function s(t) = 2t3 − 12t2 + 18t + 3, for t ≥ 0 where t is measured
in seconds and s is measured in feet.
• When is the particle at rest?
• When is the particle moving in the positive direction? negative direction?
• Find the total distance traveled in the first 5 seconds.
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Suppose a spherical balloon is being blown up so that its radius is increasing at the rate of 2 cm/s. This
means its radius after t seconds is r = 2t cm.
• Find the rate at which the volume of the sphere is changing with respect to the radius when the radius
is 3 cm.
• Find the rate at which the volume is increasing after 3 seconds. Note: The volume of a sphere is
V = 34 πr 3 .
3.4 Derivatives of Trig Functions
Important Limits:
sin x
=1
x→0 x
cos x − 1
=0
x→0
x
lim
lim
Given these limits as facts, the following must also be true: lim
x→0
Calculate the following limits.
sin 5x
x→0
x
• lim
sin 7x
x→0 sin 6x
• lim
sin 4θ
θ→0 3θ cos 7θ
• lim
8
ax
sin ax
= 1 and lim
=1
x→0 sin ax
ax
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• lim
sin 3t
4 cos 2t
• lim
cos x − 1
sin 3x
t→0
x→0
sin t
t→0 tan 2t
• lim
x2
x→0 tan2 8x
• lim
Derivatives of the Trig Functions:
d
dx
sin x = cos x
d
dx
csc x = − csc x cot x
d
dx
cos x = − sin x
d
dx
sec x = sec x tan x
d
dx
tan x = sec2 x
d
dx
cot x = − csc2 x
Calculate the derivatives of the following functions.
• f (x) = cos x + 2 sec x
• g(t) = 4t − tan t sin t
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√
3 θ cot θ
• h(θ) =
csc θ + θ 2
Find the equation of the tangent line to the graph of y = 2 sin x − 3 at the point where x = π6 .
For what values of x, 0 ≤ x < 2π, does the graph of f (x) =
10
sin x
have a horizontal tangent.
2 + cos x