In the exercise, I am required to prove that

$f'$ is odd $\iff$ $f$ is even

Moving from right to left was pretty trivial, however, I couldn't move from left to right. Note that we can only use very basic things like the definition of the derivative and rules of differentiation (No mean value theorem or any other theorems.)

I cannot even see the left implying the right, but I hope I will see it after you guys explain how. Thanks in advance.