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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

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BuyFindarrow_forward

Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

If F(x) = f(3f(4f(x))), where f(0) = 0 and f′(0) = 2, find F′(0).

To determine

To find: The value F(0).

Explanation

Given:

The function is F(x)=f(3f(4f(x))).

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(3f(4f(x))).

F(x)=ddx(F(x))=ddx(f(3f(4f(x))))

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

F(x)=f(3f(4f(x)))ddx(3f(4f(x)))

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

F(x)=f(3f(4f(x)))ddx(3f(4f(x)))=f(3f(4f(x)))[3ddx

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