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Math Help - Exponential Derivative

  1. #1
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    Exponential Derivative

    What is the derivative of

    ?

    I was thinking of converting it to x^(1/6)(ln(x)), but after I take the derivative, my answer is always wrong.
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  2. #2
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    Quote Originally Posted by Kimmy2 View Post
    What is the derivative of

    ?

    I was thinking of converting it to x^(1/6)(ln(x)), but after I take the derivative, my answer is always wrong.
    y = \left(\sqrt[6]{x}\right)^{\ln{x}}

     = \left(x^{\frac{1}{6}}\right)^{\ln{x}}

     = x^{\frac{1}{6}\ln{x}}.


    Therefore

    \ln{y} = \ln{\left(x^{\frac{1}{6}\ln{x}}\right)}

     = \frac{1}{6}\ln{(x)}\ln{(x)}

     = \frac{1}{6}\left(\ln{x}\right)^2.


    \frac{d}{dx}(\ln{y}) = \frac{d}{dx}\left[\frac{1}{6}(\ln{x})^2\right]

    \frac{d}{dy}(\ln{y})\,\frac{dy}{dx} = \frac{1}{6}\cdot \frac{1}{x}\cdot 2\ln{x}

    \frac{1}{y}\,\frac{dy}{dx} = \frac{\ln{x}}{3x}

    \frac{dy}{dx} = \frac{y\ln{x}}{3x}

    \frac{dy}{dx} = \frac{\left(\sqrt[6]{x}\right)^{\ln{x}}\ln{x}}{3x}.
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  3. #3
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    I see. I did not differentiate at all. I ended up doing something like this:

    (1/6)lnx^(lnx)

    Then

    lnx * lnx * 1/6 * lnx

    Which was completely wrong.
    Thank you so much, your explanation was so clear. I understand!
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