Chapter 3 | Derivatives
289
The Chain and Power Rules Combined We can now apply the chain rule to composite functions, but note that we often need to use it with other rules. For example, to find derivatives of functions of the form h ( x ) = ( g ( x )) n , we need to use the chain rule combined with the power rule. To do so, we can think of h ( x ) = ⎛ ⎝ g ( x ) ⎞ ⎠ n as f ⎛ ⎝ g ( x ) ⎞ ⎠ where f ( x ) = x n . Then f ′( x ) = nx n −1 . Thus, f ′ ⎛ ⎝ g ( x ) ⎞ ⎠ = n ⎛ ⎝ g ( x ) ⎞ ⎠ n −1 .
This leads us to the derivative of a power function using the chain rule, h ′( x ) = n ⎛ ⎝ g ( x ) ⎞ ⎠ n −1 g ′( x )
Rule: Power Rule for Composition of Functions For all values of x for which the derivative is defined, if h ( x ) = ⎛ ⎝ g ( x )
n .
⎞ ⎠
Then
(3.18)
⎞ ⎠ n −1 g ′( x ).
h ′( x ) = n ⎛
⎝ g ( x )
Example 3.48 Using the Chain and Power Rules
Find the derivative of h ( x ) = 1 ⎛ ⎝ 3 x 2 +1
2 .
⎞ ⎠
Solution First, rewrite h ( x ) = 1 ⎛ ⎝ 3 x 2 +1
−2
⎛ ⎝ 3 x 2 +1 ⎞ ⎠
2 =
.
⎞ ⎠
Applying the power rule with g ( x ) =3 x 2 +1, we have
−3
h ′( x ) =−2 ⎛
⎝ 3 x 2 +1 ⎞ ⎠
(6 x ).
Rewriting back to the original form gives us
h ′( x ) = −12 x
(3 x 2 +1) 3 .
⎝ 2 x 3 +2 x −1 ⎞ ⎠ 4 .
3.34
Find the derivative of h ( x ) = ⎛
Example 3.49 Using the Chain and Power Rules with a Trigonometric Function
Find the derivative of h ( x ) = sin 3 x .
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