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Math Help - Derivative Of cos(x)^ln(x) ?

  1. #1
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    Derivative Of cos(x)^ln(x) ?

    What is the derivative of cos(x)^ln(x)? When I plug it into mathematica, I don't know where the answer comes from...
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  2. #2
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    Quote Originally Posted by soma View Post
    What is the derivative of cos(x)^ln(x)? When I plug it into mathematica, I don't know where the answer comes from...
    I would approach this by taking the logarithm of both sides.

    ln(y)=ln(cos^{ln(x)}x)

    ln(y)=ln(x)lncos(x)

    The use implicit differentiation.
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  3. #3
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    So I get e^((ln(cos(x))/x)-ln(x)tan(x))
    But mathematica shows (cos(x)^ln(x))*((ln(cos(x))/x)-ln(x)tan(x))

    The difference is I have e^, while mathematica has (cos(x)^ln(x))*
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  4. #4
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    I get:

    \frac{1}{y}\frac{dy}{dx} =\frac{1}{x}lncos(x) -ln(x)tan(x)

    \frac{dy}{dx}=\frac{y}{x}lncos(x)-yln(x)tan(x)
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  5. #5
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    Oh! don't forget that y= cos^{ln(x)}(x)

    \frac{dy}{dx}=cos^{ln(x)}(x)[\frac{lncos(x)}{x}-ln(x)tan(x)]
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