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:

Another 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?