Chapter 2.6, Problem 80ES

Recall that a function $f$ is even if $f\left(-x\right)=f\left(x\right)$ and odd if $f\left(-x\right)=-f\left(x\right)$ , for all $x$ in the domain of $f$ . Assuming that $f$ is differentiable, prove:

(a) $f^{\prime }$ is odd if $f$ is even

(b) $f^{\prime }$ is even if $f$ odd.