Oh ok, I was taking the derivative of the whole thing.
When I take the first derivative I get
I'm not sure what happens to the even-ness and odd-ness of functions as you take more derivatives though.
For the first-order derivative, you won't actually need the product rule - just the chain rule. But for the second derivative on up, you will need the product rule and the chain rule, and the sum rule, etc. So no, you don't take the derivative of the whole function. Like I said, derivatives are an operator. Operators only operate on what is to their right. Example:
One more example for good measure:
And you can simplify further, but that's not the point of this example. Get the idea?
However, all of this misses the point. You need to imagine what those derivative operators are doing to the right-most (only!) exponential function. As you keep taking more and more derivatives, what happens to the even-ness of odd-ness of the overall functions?
Too many minus signs in there, I think, but that's much better.
However, what you should really be doing is taking the derivative of an arbitrary even function. An even function satisfies f(x) = f(-x) for all x. If I take the derivative of both sides of that equation, what do I get?