The Chain Rule for Functions of Two Variables


In physics and chemistry, the pressure P of a gas is related to the volume V, the number of moles of gas n, and temperature T of the gas by the following equation:

where R is a constant of proportionality. We can easily find how the pressure changes with volume and temperature by finding the partial derivatives of P with respect to V and P, respectively. But, now suppose volume and temperature are functions of time (with n constant): V=V(t) and T=T(t). We wish to know how the pressure P is changing with time. To do this we need a chain rule for functions of more than one variable. We will find that the chain rule is an essential part of the solution of any related rate problem.

The Chain Rule

If x=x(t) and y=y(t) are differentiable at t and z=f(x(t),y(t)) is differentiable at (x(t),y(t)), then z=f(x(t),y(t) is differentiable at t and

This can be proved directly from the definitions of z being differentiable at (x(t),y(t)) and x and y being differentiable at t.


For the function z(x,y)=yx^2+x+y with x(t)=log(t) and y(t)=t^2, we have


For our introductory example, we can now find dP/dt:

Implicit Differentiation

A special case of this chain rule allows us to find dy/dx for functions F(x,y)=0 that define y implicity as a function of x. Suppose x is an independent variable and y=y(x). Differentiating both sides with respect to x (and applying the chain rule to the left hand side) yields

or, after solving for dy/dx,

provided the denominator is non-zero. For example, if F(x,y)=x^2+sin(y) +y=0, then

which implies

An Extension of the Chain Rule

We may also extend the chain rule to cases when x and y are functions of two variables rather than one. Let x=x(s,t) and y=y(s,t) have first-order partial derivatives at the point (s,t) and let z=f(s,t) be differentiable at the point (x(s,t),y(s,t)). Then z has first-order partial derivatives at (s,t) with

The proof of this result is easily accomplished by holding s constant and applying the first chain rule discussed above and then repeating the process with the variable t held constant.


Let z(x,y)=x^2+y^2 with x(r,theta)=rcos(theta) and y(r,theta)=rsin(theta). The partials of z with respect to r and theta are

where in the computation of the first partial derivative we have used the identity

The Chain Rule for Functions of More than Two Variables

We may of course extend the chain rule to functions of n variables each of which is a function of m other variables. This is most easily illustrated with an example. Suppose f=f(x_1,x_2,x_3,x_4) and x_i=x_i(t_1,t_2,t_3) (i.e., we have set n=4 and m=3). Then, for example, the partial derivative of f with respect to t_2 is

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