I encountered the following expression while studying continuity equation derivation in physics.
There is a step where we go from differentiation of a volume integral with time to taking partial derivative with time inside the volume integral.
$$-\frac{\mathrm{d}Q_{in}}{\mathrm{d}t} = -\frac{\mathrm{d}}{\mathrm{d}t} { \int_{\nu} {\rho_{\nu}} \mathrm{d}\nu } {\ } {\overset{?}{=}} -\int_{\nu} {\frac{\partial {\rho_{\nu}}}{\partial t}} \mathrm{d}\nu$$
How does this work? Is this due to Leibniz Rule or Feynman's Trick in multivariable calculus?
This should be the Leibniz integral rule, yes.
Since $v$ is fixed with respect to time, the velocity of the boundary $\mathbf{\vec v}_b$ is zero, and hence the boundary term vanishes, see also link.