The derivative of e^x is e^x. This means that if f(x) = e^x, then f(x) = e^x. The proof for this can be done using the limit definition of the derivative, which involves taking the limit as h approaches 0 of (e^(x+h) - e^x)/h. Simplifying this expression using algebra and the properties of exponents, we get e^x times the limit as h approaches 0 of (e^h - 1)/h. The limit of (e^h - 1)/h is a well-known limit that approaches 1 as h approaches 0, so the derivative of e^x is e^x.