if f(x) is an even function
prove that f'(x) is odd function
Alternatively
$\displaystyle f'(x)=\lim_{\phi \to x}\frac{f(\phi)-f(x)}{\phi-x}$
So
$\displaystyle \begin{aligned}f'(-x)&=\lim_{\phi\to -x}\frac{f(\phi)-f(-x)}{\phi+x}\\
&=\lim_{\phi\to -x}\frac{f(\phi)-f(x)}{\phi+x}\\
&=\lim_{\phi\to x}\frac{f(-\phi)-f(x)}{-\phi+x}~~{\color{red}\star}\\
&=-\lim_{\phi\to x}\frac{f(\phi)-f(x)}{\phi-x}\\
&=-f'(x)\quad\blacksquare\end{aligned}$
$\displaystyle {\color{red}\star}:\phi\mapsto -\phi$