Prove the differential of a scalar multiple of a vector field

What, exactly, is the definition of this "product rule" you wish to use?

[Either you're saying it's "sort of like the product rule, but they are actually different", or the product rule you wish to use is exactly what you need here, in which case you need to prove it].

I might have a proof, but I'm not sure if it's valid. I tried writing Phi as an unknown function or x, y and z, and u as a column vector of 3 more unknown 3D functions. Then I expanded Div[(Phi)(u)] using the product rule on each term - so I had a new column vector with two terms in each direction. Is that allowed, since each entry was a function, but not a vector?

Then I rearranged things to show that the result was equivalent to the proof desired, the result being applicable to n-dimensional space since since the operations involved were all independent of number of dimensions.

Does that work? It feels like I'm assuming things I shouldn't.

In any case, just to check, the later formula in the question, is correctly interpreted as stating that the rate of change of fluid density in a given volume is equal to the rate of flow of fluid in or out of the volume, no?

Why do you have a column vector after you have "worked out" the Div. The Divergence is a scalar.

This is easiest to do with summation convention. Have you seen this before.

Then the first part would be

div($\psi$u)$= \frac{\partial}{\partial x_i}(\psi u_i)$