Chapter 3 | Derivatives
293
Then, applying the chain rule once we obtain k ′( x ) = d dx ⎛ ⎝ h ( f ⎛ ⎝ g ( x )
⎞ ⎠ · d
⎞ ⎠ ⎞ ⎠ = h ′ ⎛
⎛ ⎝ g ( x )
⎞ ⎠
f ⎛ ⎝
⎛ ⎝ g ( x )
⎞ ⎠
⎞ ⎠ .
⎝ f
dx
Applying the chain rule again, we obtain
k ′( x ) = h ′ ⎛ ⎝ f ⎛
⎞ ⎠ g ′( x ) ⎞ ⎠ .
⎞ ⎠ f ′
⎛ ⎝ g ( x )
⎝ g ( x )
Rule: Chain Rule for a Composition of Three Functions For all values of x for which the function is differentiable, if k ( x ) = h ⎛ ⎝ f ⎛ ⎝ g ( x ) ⎞ ⎠ ⎞ ⎠ ,
then
k ′( x ) = h ′ ⎛ ⎝ f ⎛
⎞ ⎠ f ′
⎞ ⎠
⎛ ⎝ g ( x )
⎞ ⎠ g ′( x ).
⎝ g ( x )
In other words, we are applying the chain rule twice.
Notice that the derivative of the composition of three functions has three parts. (Similarly, the derivative of the composition of four functions has four parts, and so on.) Also, remember, we can always work from the outside in, taking one derivative at a time. Example 3.55 Differentiating a Composite of Three Functions
Find the derivative of k ( x ) =cos 4 ⎛
⎝ 7 x 2 +1 ⎞ ⎠ .
Solution First, rewrite k ( x ) as
4
⎛ ⎝ cos
⎛ ⎝ 7 x 2 +1 ⎞ ⎠ ⎞ ⎠
k ( x ) =
.
Then apply the chain rule several times.
3 ⎛
⎞ ⎠
k ′( x ) =4 ⎛
⎛ ⎝ 7 x 2 +1 ⎞ ⎠ ⎞ ⎠ ⎛ ⎝ 7 x 2 +1 ⎞ ⎠ ⎞ ⎠ ⎛ ⎝ 7 x 2 +1 ⎞ ⎠ ⎞ ⎠
⎛ ⎝ 7 x 2 +1 ⎞ ⎠
⎝ d
⎝ cos ⎛ ⎝ cos ⎛ ⎝ cos
dx cos
Apply the chain rule.
⎛ ⎝ d
⎞ ⎠ Apply the chain rule.
3 ⎛
⎛ ⎝ 7 x 2 +1 ⎞ ⎠ ⎞ ⎠ ⎛ ⎝ 7 x 2 +1 ⎞ ⎠
⎛ ⎝ 7 x 2 +1 ⎞ ⎠
=4
⎝ −sin
dx
3 ⎛
⎞ ⎠ (14 x )
=4
⎝ −sin
Apply the chain rule.
=−56 x sin(7 x 2 +1)cos 3 (7 x 2 +1)
Simplify.
Find the derivative of h ( x ) = sin 6 ⎛ ⎝ x 3 ⎞ ⎠ .
3.38
Example 3.56
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