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.
