You can take expressions for any of the four basic 'thermodynamic potentials' and derive expressions for the others from there. For instance you can start with internal energy (U) and derive Helmoltz Free Energy (F), Gibbs Free energy (G) and Enthalpy (H).

However this can also be done by other means which are normally taught first.

So the method tends to be used for more modern (I hesitate to say more advanced) purposes involving quantum theory, statistical mechanics etc.

Here is a typical example

In a crystal where the ions have only two energy states called 0 and e the Helmholtz free energy was shown by Reidi in 1988 to be

F = NRT {1 + exp(-U/RT)}

Taking U as constant find the entropy relation and the molar heat capacity as a function of temperature.

Would working this example be of interest?

The molar heat capacity leads to a solid state effect known as the Schottky Anomaly

https://en.wikipedia.org/wiki/Schottky_anomaly