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

$${\displaystyle dx=\left({\frac {\partial x}{\partial y}}\right)dy+\left({\frac {\partial x}{\partial z}}\right)dz}$$

$${\displaystyle dy=\left({\frac {\partial y}{\partial x}}\right)dx+\left({\frac {\partial y}{\partial z}}\right)dz}$$

Substitute dy into dx

$${\displaystyle dx=\left({\frac {\partial x}{\partial y}}\right)\left[\left({\frac {\partial y}{\partial x}}\right)dx+\left({\frac {\partial y}{\partial z}}\right)dz\right]+\left({\frac {\partial x}{\partial z}}\right)dz}$$

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

$${\displaystyle \left({\frac {\partial x}{\partial y}}\right)\left({\frac {\partial y}{\partial z}}\right)+\left({\frac {\partial x}{\partial z}}\right)=0}$$

Subtracting the second term and multiplying by its inverse gives the triple product rule

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

Example: Ideal Gas Law

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

${\displaystyle PV=nRT}$

which can be written as

${\displaystyle f(P,V,T)=PV-nRT=0}$

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

${\displaystyle {\begin{aligned}P&=P(V,T)={\frac {nRT}{V}}\\[1em]V&=V(P,T)={\frac {nRT}{P}}\\[1em]T&=T(P,V)={\frac {PV}{nR}}\end{aligned}}}$

From the above expressions, we have

${\displaystyle {\begin{aligned}-1&=\left({\frac {\partial P}{\partial V}}\right)\left({\frac {\partial V}{\partial T}}\right)\left({\frac {\partial T}{\partial P}}\right)\\[1em]&=\left(-{\frac {nRT}{V^{2}}}\right)\left({\frac {nR}{P}}\right)\left({\frac {V}{nR}}\right)\\[1em]&=\left(-{\frac {nRT}{PV}}\right)\\[1em]&=-{\frac {P}{P}}=-1\end{aligned}}}$

Geometric Realization

The profile of a traveling wave at time t (solid line) and t+Δt (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

${\displaystyle \phi (x,t)=A\cos(kx-\omega t)}$

shown on the right at time t (solid blue line) and at a short time later t+Δt (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 x+Δx at time t+Δt,

${\displaystyle A\cos(kx-\omega t)=A\cos(k(x+\Delta x)-\omega (t+\Delta t)).}$

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

${\displaystyle v={\frac {\Delta x}{\Delta t}}={\frac {\omega }{k}}.}$

To elucidate the connection with the triple product rule, consider the point p₁ at time t and its corresponding point (with the same height) p̄₁ at t+Δt. Define p₂ as the point at time t whose x-coordinate matches that of p̄₁, and define p̄₂ to be the corresponding point of p₂ as shown in the figure on the right. The distance Δx between p₁ and p̄₁ is the same as the distance between p₂ and p̄₂ (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,

${\displaystyle \left({\frac {\partial \phi }{\partial t}}\right)\Delta t={\text{rise from }}p_{2}{\text{ to }}{\bar {p}}_{1}{\text{ in time }}\Delta t{\text{ (gold line)}}}$

${\displaystyle \left({\frac {\partial \phi }{\partial x}}\right)={\text{slope of the wave (red line) at time }}t.}$

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

${\displaystyle \Delta x=-{\frac {\left({\frac {\partial \phi }{\partial t}}\right)\Delta t}{\left({\frac {\partial \phi }{\partial x}}\right)}},}$

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

${\displaystyle v={\frac {\Delta x}{\Delta t}}=-{\frac {\left({\frac {\partial \phi }{\partial t}}\right)}{\left({\frac {\partial \phi }{\partial x}}\right)}}.}$

For infinitesimal Δt, ${\displaystyle {\frac {\Delta x}{\Delta t}}=\left({\frac {\partial x}{\partial t}}\right)}$ and we recover the triple product rule

${\displaystyle v={\frac {\Delta x}{\Delta t}}=-{\frac {\left({\frac {\partial \phi }{\partial t}}\right)}{\left({\frac {\partial \phi }{\partial x}}\right)}}.}$