The derivate of e^x is actually in fact e^x times the gradient at the y-intercept. Coincidently, the gradient of e^x happens to be exactly 1. Therefore the derivative of e^x is actually e^x.
Hence this is the reason why the number e is the fundamental number to apply calculus to exponential graphs. For example. With the use of logarithms, we can actually differentiate a^x, where a is any real number. We do this by using the change of base forumla. The result is 1/xlog(a). Therefore we can apply calculus to any exponential graphs.