For the composite function $$ f(x)=g(h(x)) $$, identify an inside and outside function. Write the derivative of the composite function with respect to $$ x $$. Do not use identity functions for $$ g(h) $$ or $$ h(x) $$. $$ \left\{g(x)=(-6.2-3.6 x)^{8}\right. $$ $$ f^{\prime}(x)= $$ Submit answer Next item

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Let the composite function be  
$$
f(x)=g(h(x)),
$$  
where the outside function is  
$$
g(u)=\left(-6.2-3.6u\right)^8
$$  
(with $$ u=h(x) $$) and the inside function is  
$$
h(x).
$$

To differentiate $$ f(x) $$ with respect to $$ x $$, we use the chain rule:
$$
f'(x)=g'\bigl(h(x)\bigr)\cdot h'(x).
$$

First, differentiate $$ g(u) $$ with respect to $$ u $$:
$$
g(u)=\left(-6.2-3.6 u\right)^8.
$$
Applying the power rule together with the chain rule, we get:
$$
g'(u)=8\left(-6.2-3.6u\right)^7\cdot\frac{d}{du}\left(-6.2-3.6u\right).
$$
Since
$$
\frac{d}{du}\left(-6.2-3.6u\right)=-3.6,
$$
we have:
$$
g'(u)=8\left(-6.2-3.6u\right)^7\cdot (-3.6)=-28.8\left(-6.2-3.6u\right)^7.
$$

Now substitute $$ u=h(x) $$ back into the expression for $$ g'(u) $$:
$$
g'\bigl(h(x)\bigr)=-28.8\left(-6.2-3.6h(x)\right)^7.
$$

Therefore, the derivative $$ f'(x) $$ is:
$$
f'(x)=-28.8\left(-6.2-3.6h(x)\right)^7\cdot h'(x).
$$
The derivative of the composite function $$ f(x) = g(h(x)) $$ is: $$ f'(x) = -28.8 \left(-6.2 - 3.6h(x)\right)^7 \cdot h'(x) $$

For the composite function , identify an inside and outside function. Write the derivative of the composite function with respect to . Do
not use identity functions for or .

Submit answer Next item

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For the composite function , identify an inside and outside function. Write the derivative of the composite function with respect to . Do not use identity functions for or . Submit answer Next item