Let $u$ be a function such that $|u|^{(p-2)/2}u$ is in $H^1_0(G)$, $G$ is open and $p>2$.

How can I show that

$$D(|u|^{(p-2)/2}u)=p/2|u|^{(p-2)/2}D(u) \quad(1)$$

or how can I show that, if $G$ is bounded then

$$u\in W^{1,p}_0(G) \quad (2).$$

Note that (2) implies (1).

(1) is stated in the proof of Lemma 1 in a work of Raviart "Sur la résolution et l'approximation de certaines équations paraboliques non linéaires dégénérées" published about 1967.