In class, we were asked to prove the identity $\nabla\times(\Omega\vec{V})=\Omega(\nabla\times\vec{V})-\vec{V}\times\nabla\Omega$. One of the possible approaches involved separating the del operator into two operators,
\begin{equation}\nabla\equiv\nabla_{\Omega}+\nabla_{\vec{V}}\end{equation}
where $\nabla_{\Omega}$ differentiates $\Omega$ and leaves it operates on $\vec{V}$ zero, and vice versa. I know how to prove the identity, once I accept that $\nabla\equiv\nabla_{\Omega}+\nabla_{\vec{V}}$, but I find ti difficult to convince myself that it is true. Can del be separated like this?