I'm running into a problem when trying to show the mass continuity equation for a fluid, which says
$$\frac{\partial \rho}{\partial t} + \left(\nabla \cdot \rho \textbf{u}\right) = 0$$
Where $\rho=\rho(x,y,z,t)$ is the density of the fluid and $\textbf{u} = \textbf{u}(x,y,z,t)$ is the velocity vector of the infinitesimal unit of volume [or mass].
I start by noting that $m=\iiint_V\rho dV$. Differentiating that (with chain rule) gives us
$$\begin{align*} \frac{\partial m}{\partial t} &= \frac{\partial}{\partial t}\iiint_V \rho dV\\ &= \iiint_V \frac{\partial \rho}{\partial t} dV\\ &= \iiint_V \left(\frac{\partial \rho}{\partial x}x'(t)+\frac{\partial \rho}{\partial y}y'(t)+\frac{\partial \rho}{\partial z}z'(t)+\rho_t\right)dV\\ &=\iiint_V(\nabla \rho \cdot \textbf{u} + \rho_t)dV\\ \end{align*} $$
And as you can see, that doesn't really match up with the originally stated continuity equation. I have $(\nabla \rho \cdot \textbf{u})$ instead of $(\nabla \cdot \rho \textbf{u})$. Where have I gone wrong? What needs to be corrected?