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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270336

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BuyFindarrow_forward

Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270336
Textbook Problem

Recall that a function f is called even if f(x) = f(x) for all x in its domain and odd if f(– x) = –f(x) for all such x. Prove each of the following.

(a) The derivative of an even function is an odd function.

(b) The derivative of an odd function is an even function.

(a)

To determine

To prove: The derivative of an even function is odd.

Explanation

Definition used:

The function f(x) is an even function if and only if f(x)=f(x) and the odd function if and only if f(x)=f(x).

Proof:

Assume f(x) is an even function.

This implies that, write f(x)=f(x).

Take the derivative of the function f(x)=f(x),

f(x) =limh0f(x+h)f(x)h=limh0f((xh))f(x)h

Since f(x) is an even, which implies that f(x)=f(x) and f((xh))=f(xh)

(b)

To determine

To Prove: The derivative of an odd function is even.

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