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Math Help - Exponential function

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    Exponential function

    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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    Quote Originally Posted by matty888 View Post
    Define the exponential function in terms of the natural logarithm and show that the derivitave of e^x is e^x
    Define \ln x = \int_1^x \frac{d\mu}{\mu}, for x>0. This function is differenciable by the fundamental theorem for any x>0 and \ln x = \frac{1}{x}. Since the derivative is positive the function is increasing, so \ln x is a one-to-one function on \mathbb{R}^+. Let \exp (x) be its inverse function on [tex]D[tex] (where D is the range of \ln x, it happens to be that D=\mathbb{R} but it does not matter here). Thus, \ln (\exp (x)) = x. Using the chain rule we get  \left( \exp x \right) ' \frac{1}{\exp x} = 1 \implies \left( \exp x \right) ' = \exp x.

    We can go further and define some additional properties. We know that there is a unique number e such that \ln e = 1 by intermediate value theorem. Let q be any rational number. We can show that \exp ( q ) = e^q where e^q is ordinary exponentiation of rational exponents (try provong this). Thus, it is reasonable to define e^r = \exp (r) for any real number r.
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