odd function prove

**transgalactic** · January 21st 2009, 01:27 PM

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

**Krizalid** · January 21st 2009, 01:44 PM

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

**ursa** · January 21st 2009, 01:56 PM

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

**Krizalid** · January 21st 2009, 01:59 PM

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

**Mathstud28** · January 21st 2009, 04:24 PM

Alternatively

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

So

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

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

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