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 $$ D(|u|^{(p-2)/2}u)=p/2|u|^{(p-2)/2}D(u) \label{1}\tag{1} $$ or how can I show that, if $G$ is bounded then $$ u\in W^{1,p}_0(G) \label{2}\tag{2}\;? $$ Note that \eqref{2} implies \eqref{1}.

\eqref{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", Archive for Rational Mechanics and Analysis, 25, 64-80 (1967), MR215544, Zbl 0153.42202.