Partial Derivative

Why don't you write out the equivalent equation, replacing the del operator with the partial derivatives incorporated in its meaning?

 

Well, that's not the same problem now! In your first post you stated that v was a function of both x and t and did not say anything about [itex]\rho[/itex]. Assuming the most general case, that both v and [itex]\rho[/itex] are functions of both x and t, and writing v= f(x,t)i+ g(x,t)j+ h(x,t)k then we have [itex]\rho v= \rho(x,t)f(x,t)i+ \rho(x,t)g(x,t)j+ \rho(x,t)h(x,t)k[/itex] and
[tex]\nabla\cdot \rho v= \frac{\partial \rho f}{\partial x}+ \frac{\partial\rho g}{\partial y}+ \frac{\partial \rho h}{\partial z}[/tex]
[tex]= \frac{\partial\rho}{\partial x}f+ \rho \frac{\partial f}{\partial x}+ \frac{\partial\rho}{\partial y}g+ \rho \frac{\partial g}{\partial y}+ \frac{\partial\rho}{\partial z}h+ \rho\frac{\partial h}{\partial z}[/tex]
[tex]= \nabla\rho\cdot v+ \rho \nabla\cdot v[/tex]