I will try to help you by examples
1) x^4+1 is an even function
f(-x)=(-x^4)+1=x^4+1=f(x)
the derivative of this function is
f'(x)=4x^3
f'(-x)=4(-x^3)
=-4x^3
so ,it became an odd function

2) x^3+1 is an odd function
f(-x)=(-x^3)+1=-x^3+1
the derivative of this function is
f'(x)=3x^2
f'(-x)=3(-x^2)
=3x^2=f(x)
so,it became an even function