The formula $d(pV)=pdV+Vdp$ is just Leibniz's rule on the differential quotient of the product of two function, $(fg)'=gf'+fg'$, written in the language of infinitesimals (differentials).
For one variable functions, $\frac{d(f(x)g(x))}{dx}=g(x)\frac{df(x)}{dx}+f(x)\frac{dg(x)}{dx}$.
For a function of several variables, $x_1, x_2, ..x_n$, along a curve $x_1(t), x_2(t),...x_n(t)$, the total derivative of $f^*(t)=f(x_1(t),x_2(t),..,x_n(t))$ is defined by
$$\frac{df^*(t)}{dt} = \frac{\partial f}{\partial x_1}\frac{\partial x_1}{dt}+\frac{\partial f}{\partial x_2}\frac{dx_2}{dt}+...+\frac{\partial f}{\partial x_n}\frac{dx_n}{dt},$$ and again the identity holds for $\frac{d(f^*(t)g^*(t))}{dt}=g^*(t)\frac{df^*(t)}{dt}+f^*(t)\frac{dg^*(t)}{dt}$.
Now "simplify" your notation and with Leibniz (or Euler) drop $dt$ without losing any sleep over it and write
$${d(f^*(t)g^*(t))}=g^*(t){df^*(t)}+f^*(t){dg^*(t)}$$