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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) =
√
3
4
√
− x9
5
x
1
+ 5πx2/3
2x3
1
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• g(s) =
s3 + s
s2 − 3
√
t3 − 3t2 + t t
√
• g(t) =
t
√
2
Example: An object moving in a straight line has position function f (t) = 8t t − 2t3 − 2 where position
t
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
x7 f (x)
2g(x)
3
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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: 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)?
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Example: Where is the function below not differentiable? Find f ′ (x).


−2x − 1


 2
x
f (x) =
 3x − 2


 2
x − 5x
if
if
if
if
x < −1
−1≤x≤2
2<x<4
x≥4
For what values of b and c is the following function differentiable everywhere?
f (x) =
(
x2
bx3 + c
if x ≤ 2
if x > 2
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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 3t
t→0 4 cos 2t
• lim
sin 7θ
θ→0 sin 6θ
• lim
sin 4h
h→0 3h cos 7h
• lim
cos x − 1
x→0 sin 3x
• lim
6
ax
sin ax
= 1 and lim
=1
x→0 sin ax
ax
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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
Use the definition of the derivative and the identity sin(a + b) = sin a sin b + cos a cos b to show that
d
dx sin x = cos x.
Calculate the derivatives of the following functions.
• f (x) = 2 sec x + x cos x
• g(t) = 4t − tan t sin t
• h(θ) =
cot θ
csc θ + θ 2
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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) =
8
sin x
have a horizontal tangent.
2 + cos x