First Problem

For an infinitesimal volume, informally show that

$$\frac{D}{D t}(\rho\phi\Delta V) = \rho \Delta V \frac{D \phi}{D t},$$

where $\phi$ is some differentiable propery of fluid. Hence informally deduce that

$$\frac{D}{D t} \int_V \rho \phi d V = \int_V \rho \frac{D\phi}{D t} d V.$$