Triple product rule

The triple product rule, known variously as the cyclic chain rule, cyclic relation, or Euler's chain rule, is a formula which relates partial derivatives of three interdependent variables. The rule finds application in pressure, and volume in this manner. The triple product rule for such interrelated variables x, y, and z comes from using a reciprocity relation on the result of the implicit function theorem in two variables and is given by

$\left(\frac{\partial x}{\partial y}\right)_z\left(\frac{\partial y}{\partial z}\right)_x\left(\frac{\partial z}{\partial x}\right)_y = -1$ .

Note: The third variable is considered to be an implicit function of the other two.