Use the Chain Rule to prove the following.
(a) The derivative of an even function is an odd function.
(b) The derivative of an odd function is an even function.
I can easily prove these using the definition of a derivative, but I'm having trouble showing them using the chain rule.
(a) $f(x)$ is even $ \therefore f(-x) = f(x)$.
We need to show that $f'(-x) = -f'(x)$.
Let $u = -x$.
My reasoning for the next step is that if we want to find the derivative of $f$ with respect to $x$ from $f(u)$ we need to use the chain rule and first find the derivative of $f$ with respect to $u$ and multiply that with the derivative of $u$ with respect to $x$.
$f'(x) = f'(u) \cdot u' = - f'(u) = -f'(-x)$.
$f'(x) = -f'(-x)$.
$f'(-x) = -f'(x)$.
The problem with this is that I never used the fact that $f$ is even and I feel like there is a mistake in my approach, most likely where I explained my reasoning. I just can't see it. Can you please point out the mistake and point me to the right direction?