# 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) DerivativeOfEvenoddFunctionproof 2013-03-22 13:37:57 2013-03-22 13:37:57 mathcam (2727) mathcam (2727) 5 mathcam (2727) Proof msc 26A06