Q:

$f\colon \mathbb R\to \mathbb R$, prove the implication.

First question: If $f\colon \mathbb R\to \mathbb R$ is differentiable, how would I prove the implication:
$f$ is an odd function $\Rightarrow$ $f \mathrm '$ is an even function?
Also (aka. second question), is the implication "$f \mathrm '$ is an even function $\Rightarrow f$ is an odd function" true?
I tried to solve it by picking a random odd function and find a derivative out of it, but I figured it's not really the most efficient or correct way of solving it.
Any ideas?

A:

To see the second implication fails, let $f(x)= x + 1.$

