How can I prove that the derivative of an even function is odd and vice versa with the chain rule?

Answer 1

Let f be an even function.
Then f(-x)= f(x)
Take the derivative of both sides to get
f'(-x)(-1) = f'(x)
f'(-x) = -f'(x)
By definition f' is odd

Let g be an odd function.
Then g(-x) = -g(x)
Take the derivative to get
g'(-x)(-1) = -g'(x)
g'(-x) = g'(x)
By definition f' is even

Answer 2

You need to show first that an even function of x either consists of even powers of x, or can be expressed as such using Taylor's expansion. Also that an odd function consists of odd powers. If the series are uniformly convergent you can differentiate them term by term and differentiating even powers gives odd powers and vice versa.