   Chapter 3.4, Problem 69E

Chapter
Section
Textbook Problem

Suppose f is differentiable on ℝ . Let F(x) = f(ex) and G(x) = ef(x). Find expressions for (a) F′(x) and (b) G′(x).

(a)

To determine

To find: The derivative of F(x).

Explanation

Given:

The function is F(x)=f(ex).

Result used: Chain Rule:

If h is differentiable at x and g is differentiable at h(x), then the composite function F=gh defined by F(x)=g(h(x)) is differentiable at x and F is given by the product,

F(x)=g(h(x))h(x) (1)

Calculation:

Obtain the derivative of F(x)=f(ex).

F(x)=ddx(F(x))=ddx(f(ex))

Let h(x)=ex and g(u)=f(u) where u=h(x).

Apply the chain rule as shown in equation (1),

F(x)=g(h(x))h(x) (2)

The derivative of g(h(x)) is computed as follows,

g(h(x))=g(u)=ddu(g(u))=d

(b)

To determine

To find: The derivative of G(x).

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