A Calc Proof on Derivatives?

rafeeki92

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

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

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

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

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