I want to show the following exercise:
Assume $\Omega \subset \mathbb{R}$ open and bounded,$1\leq p \leq \infty$, $f:\mathbb{R}\longrightarrow \mathbb{R}$ and $f'$ is bounded. Show if $u \in W^{1,p}(\Omega)$ then $v:= f(u)\in W^{1,p}(\Omega)$ with $\partial_iv=f'(u)\partial_iu.$
To show that $v \in W^{1,p}(\Omega)$ ,i have to show that v is in $L^p(\Omega)$ and also that the weak derivative exists and belong to $L^p(\Omega)$ for $i=1,...,n$.
That $\partial_iv=f'(u)\partial_iu$ is because $f$ is $C^1$ and $u$ belongs to $W^{1,p}(\Omega)$ so i can apply the chain rule. Is this correct ?
Can someone help me to prove that $v \in W^{1,p}(\Omega)$?
Thanks in advance