derivative of even/odd function (proof)

Suppose $f(x)=\pm f(-x)$ . We need to show that $f^{\prime}(x)=\mp f^{\prime}(-x)$ . To do this, let us define the auxiliary function $m:\mathbb{R}\to\mathbb{R}$ , $m(x)=-x$ . The condition on $f$ is then $f(x)=\pm(f\circ m)(x)$ . Using the chain rule, we have that

$\displaystyle f^{\prime}(x)$	$\displaystyle=$	$\displaystyle\pm(f\circ m)^{\prime}(x)$
	$\displaystyle=$	$\displaystyle\pm f^{\prime}\big{(}m(x)\big{)}m^{\prime}(x)$
	$\displaystyle=$	$\displaystyle\mp f^{\prime}(-x),$

and the claim follows. $\Box$

Title	derivative of even/odd function (proof)
Canonical name	DerivativeOfEvenoddFunctionproof
Date of creation	2013-03-22 13:37:57
Last modified on	2013-03-22 13:37:57
Owner	mathcam (2727)
Last modified by	mathcam (2727)
Numerical id	5
Author	mathcam (2727)
Entry type	Proof
Classification	msc 26A06