Proving a property of vector fields I have been asked to prove the following property for the vector fields$\vec F$ and $\vec G$, and scalar functions $f$ and $g$. I did a question similar to this previously, but I'm still not understanding entirely how to separate this out, and what basic equations I should be relying on to prove this.
$$\nabla \cdot(f \nabla g) = f \nabla \cdot \nabla g + \nabla  f \cdot \nabla g$$
I know that I should probably expand the left hand side of the equation, and that $\text{Div} (\nabla f)$ is equal to $\nabla \cdot \nabla f$. I think maybe that'll be needed when expanding the LHS?
Any help would be greatly appreciated. 

$$\begin{align}
\nabla \cdot (f\nabla g)&=\left(\sum_i\hat x_i\frac{\partial }{\partial x_i}\right)\cdot \left(f\sum_j\hat x_j\frac{\partial g}{\partial x_j}\right)\\\\
&=\sum_i\sum_j(\hat x_i\cdot \hat x_j)\frac{\partial }{\partial x_i}\left(f\frac{\partial g}{\partial x_i}\right)\tag 1\\\\
&=\sum_i\left(f\frac{\partial^2 g}{\partial x_i^2}+\frac{\partial f}{\partial x_i}\frac{\partial g}{\partial x_i}\right)\tag 2\\\\
&\bbox[5px,border:2px solid #C0A000]{=f\nabla \cdot \nabla g+\nabla f\cdot \nabla g}
\end{align}$$
In going from $(1)$ to $(2)$ we used both the sifting property of $\hat x_i\cdot \hat x_j=\delta_{ij}$, where $\delta$ is the Kronecker Delta, and product rule differentiation.
