Consider the ideal gas law. Let's say we want to find out how the product of pressure and volume change with respect to time. We can caluclate that as so.
$PV = nRT$
$\frac{\partial( PV)}{\partial T} = nR$
However, now say we want to figure out how pressure changes with respect to temperature. Can we then do this:
$\frac{\partial( PV)}{\partial T} = nR$
$\frac{\partial( P)}{\partial T} = \frac{nR}{V}$
If we can do this, are we allowed to play with variables in PDE in general?
Abstractly, your question appears to be whether it is valid to factor out a multiplied term from under some differential operator as $D\left(f\cdot g\right)=f\cdot D(g)$. However, your question seems confused and you seem to have mixed up variables for time and temperature.
The answer to this is that no, it is not generally valid, as we see by the product rule. See Q1347040. For ordinary derivatives, we should have $\frac{d}{dx}\left(f\cdot g\right)=f\cdot \frac{dg}{dx}+g\cdot\frac{df}{dx}$. If your volume term $V$ were a nonconstant function of temperature $T$, which it undoubtedly would be in many practical settings, you could not neglect its derivative.