Define the exponential function in terms of the natural logarithm and show that the derivitave of e^x is e^x
Define , for . This function is differenciable by the fundamental theorem for any and . Since the derivative is positive the function is increasing, so is a one-to-one function on . Let be its inverse function on [tex]D[tex] (where is the range of , it happens to be that but it does not matter here). Thus, . Using the chain rule we get .
We can go further and define some additional properties. We know that there is a unique number such that by intermediate value theorem. Let be any rational number. We can show that where is ordinary exponentiation of rational exponents (try provong this). Thus, it is reasonable to define for any real number .