# Suppose f is differentiable on R. Let F(x) = f(e^x) and G(x) = e^f(x). Find expressions for (a)...

## Question:

Suppose f is differentiable on {eq}\mathbb{R} {/eq}. Let {eq}F\left( x \right)=f\left( {{e}^{x}} \right) {/eq} and {eq}G\left( x \right)={{e}^{f\left( x \right)}} {/eq}. Find expressions for

(a) {eq}F'(x) {/eq}

(b) {eq}G'(x) {/eq}

## Differentiability:

The differentiability of the exponential function having some function in the exponent is obtained using the chain rule by first differentiating the exponential function and then the exponent function; that is, {eq}\dfrac{d}{{dx}}\left( {{e^{h\left( x \right)}}} \right) = {e^{h\left( x \right)}}\dfrac{d}{{dx}}\left( {h\left( x \right)} \right){/eq}.

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(a)

Given:

• The functions are {eq}F\left( x \right) = f\left( {{e^x}} \right){/eq}, {eq}G\left( x \right) = {e^{f\left( x \right)}}{/eq}, and...