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Math Help - Differentiation

  1. #1
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    Differentiation

    Differentiate x^{\sin x}
    I know this has got something to do with the formula \frac{d}{dx}(a^x)=(a^x)(\ln a) but i don't know how to do it. I tried using the chain rule but couldn't even figure it out. I need some pointers on how to do it.
    Thanks for any help
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  2. #2
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     x^{\sin(x)} = exp[\sin(x) \ln(x)]

    to differentiate it . Let  u = \sin(x) \ln(x)

     D(e^u) = e^u D( u )

    and  D (u) = \ln(x) D(\sin(x)) + \sin(x) D (\ln(x))

    what's next?
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  3. #3
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    so I have:
    \frac{d}{dx}(e^u)=e^u(\ln x\cos x+\frac{\sin x}{x})=e^{\sin x\ln x}(\ln x\cos x+\frac{\sin x}{x})
    But this is not the answer. I'm supposed to give x^{\sin x}(\ln x\cos x+\frac{\sin x}{x}) so i don't know how the \ln x disappears.
    Thanks
    Last edited by arze; August 19th 2009 at 01:06 AM. Reason: wrong equation
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  4. #4
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    Oh! ok, I see it now. e^{\ln x}=x Am I right?
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  5. #5
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    Yep..
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