# Math Help - power mean derivative

1. ## 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!!!

2. Originally Posted by realpart1/2
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