The Chain Rule for Partial Derivatives

When evaluating the derivative of composite functions of several variables, the chain rule for partial derivatives is often used. In this lesson, we use examples to explore this method.
Derivatives
The temperature outside depends on the time of day and the seasonal month, but the season depends on where we are on the planet. January is winter in the northern hemisphere but summer in the southern hemisphere.

When calculating the rate of change of a variable, we use the derivative. When the variable depends on other variables which depend on other variables, the derivative evaluation is best done using the chain rule for partial derivatives. In this lesson we use examples to explore two cases.

Case 1
For case 1, we’ll use:

z = z(x, y)

x = x(u), and

y = y(u)

Let’s say z depends on both x and y. We could write z = z(x, y) using function notation. We interpret this as z depends on x and y. To be specific, let z = x2y.

This gets even more interesting when x and/or y also have a functional dependence. This function of a function is sometimes referred to as a composite function. Dependence on the variable u: x = x(u) and y = y(u). Let x = u2 and y = u.

As an application, z might be the temperature at (x, y) where both x and y move with time u.

Dependency Graphs
In the diagram, under the z are x and y. There is a line joining z to x and z to y. Under both x and y a line connects to the variable u. This type of drawing is called a dependency graph.

partial_z_wrt_x_path

What is the rate of change of z with respect to u? Consider the two paths from z to u. The blue path on the left passing through x expressed as:

partial_derivatives_on_the_x_path

and the green path on the right passing through y

partial_z_wrt_y_path

expressed as:

partial_derivatives_on_the_y_path

Together these two paths add to give the total rate of change of z with respect to u:

rate_of_change_of_z_wrt_u

Partial Derivatives
Note the two formats for writing the derivative: the d and the ∂. When the dependency is one variable, use the d, as with x and y which depend only on u. The ∂ is a partial derivative, which is a derivative where the variable of differentiation is indicated and other variables are held constant. For z = x2y, the partial derivative of z with respect to x is 2xy (y is held constant). A short way to write partial derivatives is (partial z, partial x). Thus, (partial z, partial y) is x2. Why do we use a (dz, du) instead of (partial z, partial u)? At the level of u in the dependency graph, there are no other variables to hold constant.

Tabulating:

(partial z, partial x) is 2xy
(partial z, partial y) is x2
(dx, du) is 2u
(dy, du) is 1
Substitute into (dz, du):

rate_of_change_of_z_wrt_u_substitution

Chain Rule of Partial Derivatives
See how the x and y were replaced by u2 and u?

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