More about delta functions

Q: How do you define its derivative? 

A: Think of the delta function as an infinite sum of continuous (and continuously differentiable) functions that limit to the 0 or 1 when integrated monstrosity. Then the derivative of the delta function is just the infinite sum of derivatives of the functions that make up the delta function.

Q: Does the delta function have a unique series representation? That is, is there only one set of functions that one must use to construct a delta function?

A: No. The delta function does not have a unique representation. Here are several ways to construct the delta function.

reference: mathworld.wolfram

Q: Why is the integral of the derivative of the delta function equal to zero, even when the delta spike is within the limits of integration?

A: Fundamental theorem of calculus says the integral equals the arithmetic difference between the anti-derivative evaluated at the limits of integration. Since the anti-derivative equals the delta function which equals zero everywhere except a spike at one point, and the limits of integration don't fall exactly on the delta spike, the resulting term equals zero.

Q: How do you use the derivative of the delta function in integrals?

A: It picks out the negative of a function's derivative at the point zero.

Q: Is the delta function definition of a functional derivative mathematically rigorous?

A: No. In general, if a math definition was made by a physicist, it's probably not!

Wikipedia