Prove if f(x) is defined on -L< x < L, and if f(x) is odd function on (-L,L), the f'(x) is even function. and vise versa.
Using Fourier series expansion, f(x) is odd function on (-L,L) can be represented by Fourier sine series expansion. If f(x) is even function on (-L,L), the it can be represented by cosine series.
The Attempt at a Solution
By series expansion of odd and even function, if f(x) is odd and represented by sine series, then the derivative of the sine series is cosine series and it become an even function. And the reverse is true from even to odd by derivative.
Anyone have a better way of proving this than what I have instead of refering to series expansion?