Sobolev chain rule
= Sobolev chain rule
{c}
{title2=$D(\Phi(u))=\Phi^{\prime}(u)Du$}
If $u\in W^{1,2}_{\mathrm{loc}}$ and $\Phi$ is continuously differentiable with bounded derivative on the range in use, then $D(\Phi(u))=\Phi^{\prime}(u)Du$ <almost everywhere>. One proves this first for <smooth functions> and then uses <density of smooth functions in a Sobolev space> and boundedness of $\Phi^{\prime}$. On compact sets one may truncate $\Phi$ away from the range of a bounded $u$.