2012/006) prove that the derivative of an odd function is even and vice versa

1)

because the function is odd

f(-x) = - f(x)

differentiate both sides wrt x using chain rule

- f'(-x) = - f'(x) or f'(-x) = f'(x) and hence derivative is even function

2)
because the function is even

f(-x) = f(x)
differentiate both sides wrt x using chain rule

- f'(-x) = f'(x) or f'(-x) = - f'(x) and hence derivative is odd function