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The Chain Rule for Functions of Several Variables

In a Nut Shell: Recall that the chain rule for a function, y, of one independent variable,

x, was very important in that it enabled you to replace the derivative of a complicated

function with a product of simpler derivatives, called the “chain” rule.

Recall the Chain Rule: If u = u(x) and x = x(t), then du/dt = [du/dx][dx/dt]

In a Nut Shell: The chain rule extends to functions of more than one independent variable.

Now consider a function, w(x,y) (which has continuous, first-order derivatives)

and that x = g(t) and y = h(t) are differentiable functions. Then w is a

differentiable function of t .

Terminology:

w is the dependent variable, x and y are intermediate variables

and t is the independent variable

Suppose w = w( x,y ) has continuous first-order derivatives. Then

dw/dt = [∂w / ∂x] dx/dt + [∂w / ∂y] dy/dt (chain rule)

It can be helpful to construct a “tree diagram” to illustrate the chain rule structure.

Tree diagram for w = w(x,y) where x = x(t) and y = y(t).

It follows then that the chain rule for dw/dt is given as above by the expression:

dw/dt = [∂w/ ∂x] dx/dt + [∂w/ ∂y] dy/dt

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