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Math Help - Where did the Log come from in this problem?

  1. #1
    Member
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    Feb 2010
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    NYC
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    Where did the Log come from in this problem?

    The following Problem:

    f(x)=5^{x^2+2x}

    Holds a derivative of:

    f'(x)=(2*log(5)*x+2*log(5))*5^{x^2+2*x}

    I need to know how this log came about.
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  2. #2
    Senior Member
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    Jan 2010
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    f(x)=5^{x^2+2x}

    \ln f(x)=\ln \left(5^{x^2+2x}\right)

    \ln f(x)=(x^2+2x) \ln 5

    \frac{d}{dx}\left[\ln f(x)\right]=\frac{d}{dx}\left[(x^2+2x) \ln  5\right]

    \frac{f'(x)}{f(x)}=(2x+2) \ln  5

    f'(x)=f(x) (2x+2) \ln  5

    f'(x)= 5^{x^2+2x} (2x+2) \ln  5


    Of course, when calculating a derivative like this, you wouldn't actually go through all these steps. Instead, you can just use the well known rule:

    \frac{d}{dx} a^{f(x)} = a^{f(x)} f'(x) \ln a
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