Prove that the derivative of an even differentiable function is odd, and the derivative of an odd differentiable function is even.
Here are my workings so far.
Lets prove the derivative of an odd differentiable function is even first. Let the odd function be $f(x)$. We have $f(-x)=-f(x)$ and $\lim_{x\to a^-} f(x)=\lim_{x\to a^+}f(x)=\lim_{x\to a}f(x)$
$f'(-x)= \lim_{h\to 0} \frac {f(-x+h)-f(-x)}{h}= \lim_{h\to 0} \frac {-f(x-h)+f(x)}{h}$
And so, I am stuck. Thanks in advance. Hints are appreciated. Solutions are even more welcome!