fluid dynamics - Transport theorem derivation question

I am trying to rigorously go through some fluid mechanics proofs and theorems. I am currently going through a proof related to the transport theorem and I am having trouble with a step.

The steps in question are the following, first the variables are transformed according to:

$$\frac{d}{dt}\int_{W_t} \rho \mathbf{u} dV = \frac{d}{dt}\int_W (\rho \mathbf{u})(\phi (\mathbf{x},t),t)J(\mathbf{x},t)dV$$

where $\phi(\mathbf{x},t)$ is the trajectory function of the particle found at $\mathbf{x}$ at time $t=0$. $W$ is the volume of the fluid under consideration at $t=0$ and $W_t$ is that volume at time $t$. $J$ is the Jacobian determinant of the transformation, followed by differenciating under the integral sign (RHS)

$$\frac{\partial}{\partial t}(\rho \mathbf{u}) (\phi(\mathbf{x},t),t) = \left( \frac{D}{Dt} \rho \mathbf{u} \right) (\phi (\mathbf{x},t),t)$$

In the context of the first equation, why is the second equation true? When I try to prove it to myself I get:

$$\frac{D}{Dt}(\rho \mathbf{u}) = \frac{\partial}{\partial t}(\rho \mathbf{u}) + \mathbf{u}\cdot \nabla (\rho \mathbf{u})$$

and I don't know how to eliminate the second term. I think it has something to do with the trajectory function $\phi$ but I am not familiar enough with this idea to know how to proceed.