Answer
(a) If $f(x)$ is an even function, then $f'(x)$ is an odd function.
(b) If $f(x)$ is an odd function, then $f'(x)$ is an even function.
Work Step by Step
(a) Let $f(x)$ be an even function.
Then:
$f(x) = f(-x)$
$f'(x) = (-1)f'(-x)$
$f'(x) = -f'(-x)$
Thus $f'(x)$ is an odd function.
(b) Let $f(x)$ be an odd function.
Then:
$f(x) = -f(-x)$
$f'(x) = -f'(-x)\cdot (-1)$
$f'(x) = f'(-x)$
Thus $f'(x)$ is an even function.
