Can the Chain Rule Prove Even and Odd Symmetry in Derivatives?

[PerenialQuest]

Hello all,
This is a question on a problem set for my Calculus 1 class. Please help if you can.

Homework Statement

A function, f defined on the set of real numbers is said to have even symmetry if f(-x) = f(x) for all x, and is said to have odd symmetry if f(-x) = -f(x) for all x. Use the chain rule to verify that if a function has even symmetry, its derivative function has odd symmetry and vice versa. Note that I am asking you to show that this is true in general. You cannot simply cite examples, such as the fact that f(x) = x^2 has even symmetry and its derivative function f'(x) = 2x has odd symmetry. You must show that this relationship holds for all even and odd functions on the set of real numbers.

Homework Equations

Chain rule: [d/dx] f(g(x)) = f'(g(x)) * g'(x)

The Attempt at a Solution

I would love to attempt this, but I do not know where to begin. I see that this conjecture is clearly true in any specific example like the one mentioned above, but I do no know how to go about proving this in general. If someone can just suggest a strategy I can probably figure it out or I can at least try it and return with evidence of a concerted effort.

Thanks for the help. Richard

 

[Dick]

Pick g(x)=(-x) in the chain rule. Then try and start something from there.