Odd Functions and Their Derivatives: A Theorem

[Zaare]

It seems the derivate of an odd function (f(-x)=-f(x)) is an even function (f(-x)=f(x)), and vice versa. Is there a theroem about this?

 

[Muzza]

Suppose f is odd. We have that (f(-x))' = (-f(x))' = -f'(x). But by the chain rule, (f(-x))' = -f'(-x). Thus -f'(-x) = -f'(x) <=> f'(-x) = f'(x) <=> f' is even.

 

[Zaare]

Ah, that was easy. Thank you. :)