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Thread: logarithmic differentiation

  1. #1
    Junior Member
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    logarithmic differentiation

    Use logarithmic differentiation to find the derivative of the following equation.
    $\displaystyle y = (ln(x))^(cos(15 x))$

    in the home work i was doing it was a multiple choice question, and i ended up getting it right, but i'm not exactly sure how to get from the question to the answer

    $\displaystyle y'= ln(x)^{cos(15x)}(cos(15x)/x(ln(x))-15(sin(15x))(ln(ln(x)))$

    i worked out the problem and got everything in this answer except for the $\displaystyle (cos(15x)/x(ln(x))$ part, and i don't know where it came from in the derivative formulas that i have.

    I'm asking in case there is a question like this on the test and i'll have no idea how to do it correctly.
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  2. #2
    Senior Member Peritus's Avatar
    Joined
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    $\displaystyle y = \ln x^{\cos 15x} $

    here's a useful little trick:


    $\displaystyle \ln y = \ln \left( {\ln x} \right)^{\cos 15x} = \cos 15x\ln \ln x
    $

    now differentiate the equation:


    $\displaystyle \begin{gathered}
    \frac{{y'}}
    {y} = \ln \ln x\frac{d}
    {{dx}}\cos 15x + \cos 15x\frac{d}
    {{dx}}\ln \ln x \hfill \\
    \hfill \\
    \Leftrightarrow y' = \ln v^{\cos 15x} \left( {\cos 15x\frac{1}
    {{x\ln x}} - 15\sin 15x\ln \ln x} \right) \hfill \\
    \end{gathered}
    $
    Last edited by Peritus; Feb 26th 2008 at 10:33 AM.
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