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Math Help - Finding the derivative of an exponential using the difference quotient

  1. #1
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    Finding the derivative of an exponential using the difference quotient

    I'm trying to find how to take the derivative of 2^x by use of the difference quotient, rather than simply using derivative rules. I know it's ln(2) * 2^x, but I need to essentially prove it. Does anyone know how?
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  2. #2
    Junior Member toraj58's Avatar
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     u = 2^x

    so

     \ln(u) = \ln(2^x)

    differntiate two sides of the equation:

     (\ln(u))^\prime = (\ln(2^x))^\prime

    left is equat to :  \frac {u^\prime}{u}

    so we have:

    \frac {u^\prime}{u} = (x\ln 2)^\prime

    then we have:

    u^\prime = u(x\ln 2)^\prime

    u^\prime = 2^x(1\times\ln 2 + 0\times x)

    u^\prime = 2^x\ln 2
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  3. #3
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    But toraj58's response doesn't address the question: find the integral using the difference quotient; i.e. the basic definition of the derivative.

    pantsaregood, the difference quotient for a function f if [f(x+h)- f(x)]/h. What is that when f(x)= 2^x? You can use one of the laws of exponents to simplify that a little.
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  4. #4
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    2^x

    [2^(x+h)-2^x]/h

    That really doesn't get me anywhere, though.
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