Triple product rule


The triple product rule, known variously as the cyclic chain rule, cyclic relation, cyclical rule or Euler's chain rule, is a formula which relates partial derivatives of three interdependent variables. The rule finds application in thermodynamics, where frequently three variables can be related by a function of the form f(x, y, z) = 0, so each variable is given as an implicit function of the other two variables. For example, an equation of state for a fluid relates temperature, 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, and is given by

where each factor is a partial derivative of the variable in the numerator, considered to be a function of the other two.

The advantage of the triple product rule is that by rearranging terms, one can derive a number of substitution identities which allow one to replace partial derivatives which are difficult to analytically evaluate, experimentally measure, or integrate with quotients of partial derivatives which are easier to work with. For example,

Various other forms of the rule are present in the literature; these can be derived by permuting the variables {x, y, z}.



An informal derivation follows. Suppose that f(x, y, z) = 0. Write z as a function of x and y. Thus the total differential dz is


Suppose that we move along a curve with dz = 0, where the curve is parameterized by x. Thus y can be written in terms of x, so on this curve


Therefore, the equation for dz = 0 becomes


Since this must be true for all dx, rearranging terms gives


Dividing by the derivatives on the right hand side gives the triple product rule


Note that this proof makes many implicit assumptions regarding the existence of partial derivatives, the existence of the exact differential dz, the ability to construct a curve in some neighborhood with dz = 0, and the nonzero value of partial derivatives and their reciprocals. A formal proof based on mathematical analysis would eliminate these potential ambiguities.

Alternative derivation


Suppose a function f(x, y, z) = 0, where x, y, and z are functions of each other. Write the total differentials of the variables

Substitute dy into dx
By using the chain rule one can show the coefficient of dx on the right hand side is equal to one, thus the coefficient of dz must be zero
Subtracting the second term and multiplying by its inverse gives the triple product rule



Example: Ideal Gas Law


The ideal gas law relates the state variables of pressure (P), volume (V), and temperature (T) via


which can be written as


so each state variable can be written as an implicit function of the other state variables:


From the above expressions, we have


Geometric Realization

The profile of a traveling wave at time t (solid line) and tt (dashed line). In the time interval Δt, the point p2 will rise up to the same height that p1 had at time t.

A geometric realization of the triple product rule can be found in its close ties to the velocity of a traveling wave


shown on the right at time t (solid blue line) and at a short time later tt (dashed). The wave maintains its shape as it propagates, so that a point at position x at time t will correspond to a point at position xx at time tt,


This equation can only be satisfied for all x and t if k Δxω Δt = 0, resulting in the formula for the phase velocity


To elucidate the connection with the triple product rule, consider the point p1 at time t and its corresponding point (with the same height) 1 at tt. Define p2 as the point at time t whose x-coordinate matches that of 1, and define 2 to be the corresponding point of p2 as shown in the figure on the right. The distance Δx between p1 and 1 is the same as the distance between p2 and 2 (green lines), and dividing this distance by Δt yields the speed of the wave.

To compute Δx, consider the two partial derivatives computed at p2,


Dividing these two partial derivatives and using the definition of the slope (rise divided by run) gives us the desired formula for


where the negative sign accounts for the fact that p1 lies behind p2 relative to the wave's motion. Thus, the wave's velocity is given by


For infinitesimal Δt,   and we recover the triple product rule


See also



  • Elliott, J. R.; Lira, C. T. (1999). Introductory Chemical Engineering Thermodynamics (1st ed.). Prentice Hall. p. 184. ISBN 0-13-011386-7.
  • Carter, Ashley H. (2001). Classical and Statistical Thermodynamics. Prentice Hall. p. 392. ISBN 0-13-779208-5.