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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

Suppose f is a differentiable function such that f(g(x)) = x and f′(x) = 1 + [f(x)]2. Show that g′(x) = 1/(1 + x2).

To determine

To show: The derivative of g(x)=11+x2.

Explanation

Given:

The function is f(g(x))=x.

Result used: Chain Rule

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

F(t)=f(g(t))g(t) (1)

Calculation:

Consider f(g(x))=x.

Differentiate on both sides with respect to x.

ddx(f(g(x)))=ddx(x)f(g(x))g(x)=1 

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