odd function prove

• Jan 21st 2009, 01:27 PM
transgalactic
odd function prove
if f(x) is an even function
prove that f'(x) is odd function
• Jan 21st 2009, 01:44 PM
Krizalid
It's quite simple since $\displaystyle f(x)=-f(-x),$ now just differentiate.
• Jan 21st 2009, 01:56 PM
ursa
Quote:

It's quite simple since http://www.mathhelpforum.com/math-he...5a6b8ad8-1.gif 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
• Jan 21st 2009, 01:59 PM
Krizalid
I let myself go with the word "odd," haha; anyway, the idea is the same.
• Jan 21st 2009, 04:24 PM
Mathstud28
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}

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