How would you prove that if $\displaystyle f$ is an even function ( $\displaystyle f(-x)=f(x)$ ) and has a derivative at every point, then the derivative $\displaystyle f'$ is an odd function ( $\displaystyle f'(-x) = -f'(x)$ ) ?
How would you prove that if $\displaystyle f$ is an even function ( $\displaystyle f(-x)=f(x)$ ) and has a derivative at every point, then the derivative $\displaystyle f'$ is an odd function ( $\displaystyle f'(-x) = -f'(x)$ ) ?
Perhaps because you don't understand...?
If $\displaystyle f(-x)=f(x)$ then $\displaystyle (f(-x))'=f'(x)$ ; OTOH, applying the chain rule we have that $\displaystyle (f(-x))'=-f'(-x)$ and thus we get $\displaystyle f'(x)=-f'(-x)\Longleftrightarrow f'(-x)=-f'(x)$ ...
Tonio