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$m:ℝ\to ℝ$, $m(x)=-x$.The condition on $f$ is then $f(x)=\pm (f\circ m)(x)$.Using the chain rule, we have that

$f^{\prime }(x)$$=$$\pm {(f\circ m)}^{\prime }(x)$

$=$$\pm f^{\prime }\left(m(x)\right)m^{\prime }(x)$

$=$$\mp f^{\prime }(-x),$

and the claim follows. $\mathrm{\square }$