A Calc Proof on Derivatives?

[rafeeki92]

A function f is even if f(-x) = f(x) and function f is odd when f(-x) = -f(x), for all domain of f. Assuming f is differentiable, prove:

a) f' is odd when f is even.
b) f' is even when f is odd.


Should I use contradiction? If so...how shall I go about it?

 

[BigGlenntheHeavy]

\(\displaystyle a) \ Even \ function: \ f(-x) \ = \ f(x)\)

\(\displaystyle \frac{d[f(-x)]}{dx} \ = \ \frac{d[f(x)]}{dx}\)

\(\displaystyle f'(-x)(-1) \ = \ f'(x)(1), \ chain \ rule.\)

\(\displaystyle f'(-x) \ = \ -f'(x), \ odd \ function.\)

\(\displaystyle Do \ b \ the \ same \ way.\)