The Chain Rule for Functions of Several Variables (continued)

The concept of the tree diagram can be extended to many other cases. i.e.

Consider the case where the dependent variable, w, has two intermediate

variables, u and v, and two independent variables, x and y.

w = w(x, y) where u = u(x,y), v = v(x,y) where

w is the dependent variable, u and v are intermediate variables,

and x and y are the independent variables

The tree structure for the dependent variable, w, is as follows:

So the two partial derivatives for w(x,y) are as follows: (note “product” of partials)

∂w/∂x = [∂w/ ∂u] ∂u/∂x + [∂w/ ∂v] ∂v/∂x

and ∂w/∂y = [∂w/ ∂u] ∂u/∂y + [∂w/ ∂v] ∂v/∂y

Now consider the case where the dependent variable, w, has three intermediate

variables and two independent variables.

w = w(x, y, z) where x = x(u,v), y = y(u,v), and z = z(u,v)

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

and u and v are the independent variables (note “product” of partials)

So ∂w/∂u = [∂w/ ∂x] ∂x/∂u + [∂w/ ∂y] ∂y/∂u + [∂w/ ∂z] ∂z/∂u

and ∂w/∂v = [∂w/ ∂x] ∂x/∂v + [∂w/ ∂y] ∂y/∂v + [∂w/ ∂z] ∂z/∂v

Click here for an example.

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