I am reading this note on the Bernoulli equations with the following derivations:

I am struggling to find a calculus based meaning for the above equations involving the infinitesimal $\delta V$: I tried replacing $\delta V$ with differential form, but $\delta V$ doesn't behave like $dV$, the volume element because $\frac{d}{dt}(\rho dV)\neq 0$; I am trying to replace $\delta V$ with $\int_{\delta V}$, basically summing the quantity $\delta V$ is concatenated with over a small drop of liquid $\delta V$, but this brings the problem of how to show $\frac{d}{dt}(\frac{1}{2}\rho v^2\delta V)=\delta V\rho v\cdot\frac{dv}{dt}$ given $\frac{d}{dt}(\rho\delta V)$.

Could people help me with understanding these equations involving $\delta V$s?