Index notation proving identity

Question

I am trying to prove the identity $\nabla\cdot(\phi\textbf{u})=\phi\nabla\cdot\textbf{u} + \textbf{u}\cdot\nabla\phi$ using index notation but I am a bit stuck. I have so far written $\nabla\cdot(\phi\textbf{u})=\nabla_i(u^i\phi)$ and I am wondering whether I am allowed to just use product rule on what I have? Thanks. Edu

Answer 1

$\nabla \cdot (\phi u) = \partial_i(\phi u_i) = \phi \partial_i u_i + u_i \partial_i \phi = \phi \nabla \cdot u + u \cdot \nabla \phi $

I think that's right. But yeah, you just use the product rule.