The triple product rule, also called the cyclic chain rule or Euler's chain rule, relates partial derivatives of three interdependent variables. It is especially useful in thermodynamics, where a relation such as f(x, y, z) = 0 allows each variable to be expressed implicitly in terms of the other two. For example, an equation of state links temperature, pressure, and volume of a fluid.

By rearranging the rule one obtains substitution identities that replace difficult-to‑evaluate partial derivatives with quotients of easier ones. The rule arises from a reciprocity relation applied to the implicit function theorem and can be derived by permuting the variables {x, y, z}. Although informal proofs assume the existence of partial derivatives and non‑zero values, a rigorous analysis removes these ambiguities.

A classic illustration is the ideal gas law, which connects the state variables of pressure (P), volume (V), and temperature (T). The law can be written in the form P V = nRT, allowing each variable to be treated as an implicit function of the other two. The triple product rule then provides a convenient method for computing cross‑derivatives in this context.

Geometric realizations, such as the propagation of a traveling wave, further demonstrate the rule’s utility in relating spatial and temporal derivatives through the phase‑velocity expression.