# function odd derivative

• December 7th 2008, 01:23 AM
hoger
function odd derivative
function f: R.....R called even if f(x)=f(-x) all x belongs R .function g : R.....R called odd if g(x)= -g( -x) all x belongs R .show with help of difinition of derivative that the derivative of an even deriverbar function is an odd function .

thank you
• December 7th 2008, 11:34 PM
CaptainBlack
Quote:

Originally Posted by hoger
function f: R.....R called even if f(x)=f(-x) all x belongs R .function g : R.....R called odd if g(x)= -g( -x) all x belongs R .show with help of difinition of derivative that the derivative of an even deriverbar function is an odd function .

thank you

For an even function $f(x)$ :

$f'(-x)=\lim_{h\to 0} \frac{f(-x+h)-f(-x)}{h}=$ $\lim_{h\to 0} \frac{f(x-h)-f(x)}{h}=-\lim_{h\to 0} \frac{f(x)-f(x-h)}{h}=-f'(x)$

CB