a function f is an even function if f(-x)=f(x) for all x and is an odd function if f(-x)=-f(x) for all x. Prove that the derivative of an even function is odd and the derivative of an odd function is even

Question

hunus · Answer

even
$$f(-x)=f(x)$$
odd
$$f(-x)=-f(x)$$
$$f(x)=a_{n}x^n+a_{n-1}x^{n-1}+a_{n-2}x^{n-2}+...+a_{1}x+a_{0}$$
for even polynomial functions n must be even
for odd polynomial functions n must be odd
Given an even function of the form
$$f(x)=a_{n}x^n+a_{n-1}x^{n-1}+a_{n-2}x^{n-2}+...+a_{1}x+a_{0}$$
$$f'(x)=na_{n}x^{n-1}+(n-1)a_{n-1}x^{n-2}+(n-2)a_{n-2}x^{n-3}+...+a_{1}$$
Lowering the power of the even function 
$$n \rightarrow (n-1)$$
by one results in an odd function
etc etc