Define the exponential function in terms of the natural logarithm and show that the derivitave of e^x is e^x

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- Apr 10th 2008, 09:20 AMmatty888Exponential function
Define the exponential function in terms of the natural logarithm and show that the derivitave of e^x is e^x

- Apr 10th 2008, 09:52 AMThePerfectHacker
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 .