(See also Bridgman's thermodynamic equations for the use of exact differentials in the theory of thermodynamic equations)
Suppose we have five state functions , and . Suppose that the state space is two-dimensional and any of the five quantities are differentiable. Then by the chain rule
but also by the chain rule:
and
so that (by substituting (2) and (3) into (1)):
which implies that (by comparing (4) with (1)):
Letting  in (5) gives:
Letting  in (5) gives:
Letting  and  in (7) gives:
using ( gives the triple product rule: