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Math Help - i need some help deriving these log functions

  1. #1
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    i need some help deriving these log functions

    i looked at the steps, and the ones i found don't make sense to me. my teacher only allows us to use properties that we have learned previously for each problem
    anyways, i cant figure out how to do these at all, so if you could point me in the right direction of how to solve these in the most basic way...

    i need some help deriving these log functions:

    f(x)= (log x)^x



    f(x)=x^ log x
    Last edited by twostep08; November 19th 2009 at 07:53 AM. Reason: Included post title in main body of post
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  2. #2
    MHF Contributor chisigma's Avatar
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    a) for f(x)=x\cdot \ln x You have to apply the 'product rule' ...

    f(x)= u(x)\cdot v(x) \rightarrow f^{'}(x)= u(x)\cdot v^{'}(x) + v(x)\cdot u^{'} (x) (1)

    b) for f(x)= x^{\ln x} You have to apply the rule...

    f(x) = u\{v(x)\} \rightarrow f^{'} (x) = \frac{du}{dv} \cdot v^{'} (x) (2)

    ... remembering that...

    x^{\ln x} = e^{\ln^{2} x} (3)

    Kind regards

    \chi \sigma
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  3. #3
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    damn..im terrible sorry, but i meant to type (log x)^x for the first one
    apparently, some keys of my keyboard arent what they should be

    that one was and still is causing me heart ache
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  4. #4
    MHF Contributor chisigma's Avatar
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    All right!... for f(x)= (\ln x)^{x} You have to apply again the rule...

    f(x)= u \{v(x)\} \rightarrow f^{'}(x) = \frac{du}{dv}\cdot v^{'}(x)

    ... remembering that...

    (\ln x)^{x} = e^{x\cdot \ln \ln x}

    Kind regards

    \chi \sigma
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