It is easy to prove that
$\lim_{p\rightarrow 1} m_p = (x_1 + \cdots + x_n)/n$. The following fact about the derivative of $m_p$ with respect to $p$ is also elementary:
$$m'_p =\frac{dm_p}{dp}
=\frac{1}{p \log p}\cdot\Big[\frac{x_1p^{x_1}+\cdots+x_np^{x_n}}{p_1^{x_1}+\cdots+p_n^{x_n}}-m_p\Big].$$
My interest in this is to create an alternative to the power mean, called the *exponential mean*: see here and here. The limit I am interested in is $\lim_{p\rightarrow 1} m'_p$. Using WolframAlpha, I computed the limit for $n=2,3,4,5$ (see here) and the following remarkable pattern emerges:
$$\lim_{p\rightarrow 1} m'_p=\frac{1}{2n^2}\sum_{1\leq i<j\leq n}(x_i-x_j)^2.$$
How do you go about formally proving this fact? It does not sound elementary to me. Also, it sounds like $m_p$ is a strictly increasing function of $p$ (its derivative beeing positive everywhere, with $m'_0 =+\infty$ and $m'_\infty =0$) unless all the $x_i$'s are identical.

**Update**

In short, $m_1$ is the arithmetic mean and $m'_1$ is half the empirical variance of $x_1,\cdots,x_n$. I tried to see if such simple formulas existed for the power mean $M_p$, but I could not find anything interesting other than the well known fact that $M_1=m_1$ is the arithmetic mean. It would be interesting to see how the second and third derivatives of $m_p$ at $p=1$ are linked to the higher empirical moments of $x_1,\cdots,x_n$.