Calculus: Early Transcendentals 9th Edition

Published by Cengage Learning
ISBN 10: 1337613924
ISBN 13: 978-1-33761-392-7

Chapter 2 - Section 2.8 - The Derivative as a Function - 2.8 Exercises - Page 165: 63

Answer

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

Work Step by Step

(a) Suppose $f(x)$ is an even function. Then $f(-x) = f(x)$ for all $x$ in the domain. Suppose that $f'(a) = c$ Then: $\lim\limits_{h \to 0^+}\frac{f(a+h)-f(a)}{h} = c$ Consider $f'(-a)$: $f'(-a) = \lim\limits_{h \to 0^-}\frac{f(-a+h)-f(-a)}{h}$ $=\lim\limits_{h \to 0^+}\frac{f(-a-h)-f(-a)}{-h}$ $=-[\lim\limits_{h \to 0^+}\frac{f(a+h)-f(a)}{h}]$ $=-c$ Note that $f'(-a) = -f'(a)$ Therefore, $f'(x)$ is an odd function. The derivative of an even function is an odd function. (b) Suppose $f(x)$ is an odd function. Then $f(-x) = -f(x)$ for all $x$ in the domain. Suppose that $f'(a) = c$ Then: $\lim\limits_{h \to 0^+}\frac{f(a+h)-f(a)}{h} = c$ Consider $f'(-a)$: $f'(-a) = \lim\limits_{h \to 0^-}\frac{f(-a+h)-f(-a)}{h}$ $=\lim\limits_{h \to 0^+}\frac{f(-a-h)-f(-a)}{-h}$ $=-[\lim\limits_{h \to 0^+}\frac{-f(a+h)+f(a)}{h}]$ $=-(-c)$ $=c$ Note that $f'(-a) = f'(a)$ Therefore, $f'(x)$ is an even function. The derivative of an odd function is an even function.
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