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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

Use the Chain Rule to prove 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 show: The derivative of an even function is an odd function.

Explanation

Definition used:

(i) If f(x) even function, then f(x)=f(x).

(ii) If f(x) odd function, then f(x)=f(x).

Result used: Chain Rule

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

F(t)=g(h(t))h(t) (1)

Proof:

Let f(x) is even function.

Then, f(x)=f(x).

The derivative of f(x) is computed as follows,

f(x)=ddx(f(x))=ddx(f(x))    (Q f(x)=f(x))

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

(b)

To determine

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

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