A function f is called EVEN if f(-x)=f(x) and ODD if f(-x)=-f(x)?

Question

a) Prove that the derivative of an even function is an odd function
b) Prove tha the derivative of an odd function is an even function

Any help is apreciated!

Answer 1

you should be able to do this using difference equations. Remember that the derivative of f(x) is equal to:

lim (x==>0) [f(x+h)-f(x)]/h

But we also get:

lim (x==>0) [f(x-h)-f(x)]/(-h)


If the function is even, then we can look at f'(-x) and get:

lim (x==>0) [f(-x+h)-f(x)]/h =
lim (x==>0) [f(-(-x-h))-f(x)]/h =
lim (x==>0) [f(x-h)-f(x)]/h=
lim (x==>0) -[f(-x+h)-f(x)]/(-h)

The proof for the Odd functions is similar.

However -- there is an easier way to get it for odd functions. You can use the chain rule. Recall that:

the derivative of f(g(x)) = f'(g(x)*g'(x)

So -- the derivative of f'(-x) = f'(-x)*(-1) = -f'(x)

There may be some way to use the chain rule for even functions too.