Results 1 to 2 of 2

Thread: Where did the Log come from in this problem?

  1. #1
    Member
    Joined
    Feb 2010
    From
    NYC
    Posts
    98

    Where did the Log come from in this problem?

    The following Problem:

    $\displaystyle f(x)=5^{x^2+2x}$

    Holds a derivative of:

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

    I need to know how this log came about.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member
    Joined
    Jan 2010
    Posts
    354
    $\displaystyle f(x)=5^{x^2+2x}$

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

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

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

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

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

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


    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:

    $\displaystyle \frac{d}{dx} a^{f(x)} = a^{f(x)} f'(x) \ln a$
    Follow Math Help Forum on Facebook and Google+


/mathhelpforum @mathhelpforum