TITLE: A proof for $\nabla |u|^p = p\ \text{sgn}(u)|u|^{p-1}\nabla u$.
QUESTION [9 upvotes]: I believe the formula 
$$\nabla |u|^p = p\ \text{sgn}(u)|u|^{p-1}\nabla u
$$ 
to be true for $u\in W^{1,p}(\Omega)$, where $\Omega$ is an open domain with nice boundary, but failed to find a good reference for this.  I am sorry if this has been asked already.
The usual assumption that I usually see regarding the chain rule for Sobolev functions, i.e.
$$
\nabla F(u) = F'(u)\nabla u,
$$
is that $F\in C^1$ with bounded derivative (or perhaps a Lipschitz function). I want $F(t)=|t|^p$, which doesn't satisfy that assumption.
My idea of the proof goes as follow: I'd like to mimic the proof of the case $p=1$ so first we show a similar result for
$$
F_\varepsilon(t) = (\varepsilon^2 + t^2)^{p/2}
$$
by approximation with $F_\varepsilon(u_n)$, where $u_n\to u$ in $W^{1,p}$ and $u_n\in C_c^\infty(\Omega)$. Then we take $\varepsilon \to 0$ to approximate $|\cdot|^p$. While the usual argument using the Dominated Convergence Theorem wouldn't work, I think Vitali Convergence Theorem might do the trick.

Does anyone know a reference to this result? Also, if the formula doesn't hold in general spaces (i.e. $u\notin W^{1,p}$) then does anyone have a counterexample?

I have seen the proofs of the cases $p=1,2$. Finding a reference for a general $p\in [1,\infty)$ turns out to be a lot harder than I anticipated.

REPLY [2 votes]: A nice proof in Lieb and Loss's Analysis Chapter 6 (Distributions)  page 152.