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Math Help - Differentiating with trig funtions

  1. #1
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    Differentiating with trig funtions

    How does one go about differentiating x^cos(x) ?

    tried doing it using chain rule. not sure though.
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  2. #2
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by pseizure2000 View Post
    How does one go about differentiating x^cos(x) ?

    tried doing it using chain rule. not sure though.
    Let y = x^{cos(x)}

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

    Now take the derivative:
    \frac{1}{y} \frac{dy}{dx} = -sin(x)~ln(x) + cos(x) \cdot \frac{1}{x}

    So
    \frac{dy}{dx} = y \left ( \frac{cos(x)}{x} - sin(x)~ln(x) \right )

    and we know what y is so,
    \frac{dy}{dx} =  (x^{cos(x)} ) \left ( \frac{cos(x)}{x} - sin(x)~ln(x) \right )

    -Dan
    Last edited by topsquark; October 19th 2007 at 11:24 AM. Reason: Oopsies!
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    why is it negative x^cos(x) in the final answer?
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  4. #4
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by pseizure2000 View Post
    why is it negative x^cos(x) in the final answer?
    It isn't negative. When I typed it in the first time I had factored the negative out, but on second thought I decided not to. I evidently forgot to erase the negative sign in the final answer.

    Sorry! (I have fixed the problem in the original post.)

    -Dan
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