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Math Help - power mean derivative

  1. #1
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    Exclamation power mean derivative

    For given positive values a_{1},\ldots,a_{n}, I have the power mean: P(\alpha):=\left(\frac{1}{n}\sum_{k=1}^{n}a_{k}^{\  alpha}\right)^{\frac{1}{\alpha}} for which I need to find \frac{d}{d\alpha}P(\alpha). Please Help!!!
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by realpart1/2 View Post
    For given positive values a_{1},\ldots,a_{n}, I have the power mean: P(\alpha):=\left(\frac{1}{n}\sum_{k=1}^{n}a_{k}^{\  alpha}\right)^{\frac{1}{\alpha}} for which I need to find \frac{d}{d\alpha}P(\alpha). Please Help!!!
    The simplest way to do this is probably as follows:

    [P(\alpha)]^{\alpha}=\frac{1}{n}\sum a_i^{\alpha}

    or:

    e^{\alpha \ln(P(\alpha))}=\frac{1}{n}\sum  e^{\alpha\ln(a_i)}

    Now differentiate this with respect to \alpha and rearrange and simplify.

    CB
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