How does one think in terms of thermodynamic differentials?

Great question! I remember being mystified with the elusive nature of thermodynamics and its formalism. But the fact of the matter is that like most (if not all) of physics, at the core of it, they’re just differential equations.

To look at this the first and fundamental thing to note is that thermodynamics is a study of the interplay between macroscopic quantities such as energy $E$, volume $V$, pressure $P$, temperature $T$ and something called as entropy $S$. There could be systems with more quantities such as electric potential $V$ and such but in most cases it’s limited to mechanical systems where only the quantities mentioned earlier are relevant.

These quantities aren’t independent of each other and follow a constraint equation which is called as the equation of state, generally expressed in the form $E(S,V)$ Why does the relation not contain our other variables like $T,P$? We’ll find out soon enough. But before we go there one must realise that once we have an all encompassing relation between the physical quantities, we should be able to uniquely define every possible state our system can take.

Now we are interested in asking if there’s a tiny change in one of our quantities, what’ll happen to the state of our system. Why do we care about this? Because this way we can find a way to go from one state to another. For example the room is cold. And we want to warm it up. But we can control only pressure (say) then how do we go about it? Our equation of state contains all the information we need! $$dE(S,V)=\frac{\partial E}{\partial S}dS + \frac{\partial E}{\partial V}dV$$

Turns out the partial derivatives of energy turn out to be our temperature (wrt S) and negative pressure (wrt V)! The reason for this can be deduced by considering how the system equilibriates under different constraints. So finally we have, from our equation of state, it’s differential form: $$dE=TdS-PdV$$

The reason why the expression has this form is because the different variables aren’t independent of each other. They obey a constraint equation. And most importantly, it’s a multivariate constraint. Take Newton’s second law for example, can be expressed in the following form: $$Fdt=mdv$$ But there’s a more convenient way to express it, namely $F=ma$. Unfortunately in the thermodynamic case, due to the reasons listed earlier, this isn’t possible and the most reasonable way to express it is in terms of differentials.

Hope this helps!