Thread: odd function prove

if f(x) is an even function
prove that f'(x) is odd function

It's quite simple since $\displaystyle f(x)=-f(-x),$ now just differentiate.

It's quite simple since now just differentiate.

krizalid you make a mistake
its an even function
so
f(-x)=f(x), now differentiate
-f'(-x)=f'(x)
f'(-x)=-f'(x)
hence, f'(x) is an odd function

I let myself go with the word "odd," haha; anyway, the idea is the same.

Alternatively

$\displaystyle f'(x)=\lim_{\phi \to x}\frac{f(\phi)-f(x)}{\phi-x}$

So

$\displaystyle \begin{aligned}f'(-x)&=\lim_{\phi\to -x}\frac{f(\phi)-f(-x)}{\phi+x}\\
&=\lim_{\phi\to -x}\frac{f(\phi)-f(x)}{\phi+x}\\
&=\lim_{\phi\to x}\frac{f(-\phi)-f(x)}{-\phi+x}~~{\color{red}\star}\\
&=-\lim_{\phi\to x}\frac{f(\phi)-f(x)}{\phi-x}\\
&=-f'(x)\quad\blacksquare\end{aligned}$

$\displaystyle {\color{red}\star}:\phi\mapsto -\phi$