Homework Soursea Prove that the derivative of every even function is an odd
a) Prove that the derivative of every even function is an odd function
b) Prove that the derivative of every odd function is an even function
Solve both using the chain rule
b) Prove that the derivative of every odd function is an even function
Solve both using the chain rule
Solution
a) If f is even f(x)=f(-x) so, by the chain rule f\'(x)=-f\'(x) Thus f\' is odd. b) Since f is odd, then f (-x) = - f (x) f \' (-x)= f \'(-x) (-1) = -f \'(x) (-1) since f is odd (this is the application of chain rule) = f \' (x) which is an even function.
