Let $P:\mathbb{R}^n\rightarrow\mathbb{R}$ be the homogeneous polynomial of degree $k$:
$$P(x)=\sum_{|a|=k}c_{\alpha}x^{\alpha}$$
How can I show:
$\partial^{\beta}P(x)=\beta !c_{\beta}$ for all $x\in\mathbb{R}^n$ and all multi-indices $\beta\in\mathbb{N}^n$ with $|\beta |=k$?
Hint: If $\alpha \ne \beta$ with $|\alpha| = k$, some index $i$ has $\alpha_i < \beta_i$, and then $\partial^\beta x^\alpha = 0$. Then consider the case of $\partial^\beta x^\beta$.