Differentiation of Functions of Several Variables

# 25 The Chain Rule

### Learning Objectives

• State the chain rules for one or two independent variables.
• Use tree diagrams as an aid to understanding the chain rule for several independent and intermediate variables.
• Perform implicit differentiation of a function of two or more variables.

In single-variable calculus, we found that one of the most useful differentiation rules is the chain rule, which allows us to find the derivative of the composition of two functions. The same thing is true for multivariable calculus, but this time we have to deal with more than one form of the chain rule. In this section, we study extensions of the chain rule and learn how to take derivatives of compositions of functions of more than one variable.

### Chain Rules for One or Two Independent Variables

Recall that the chain rule for the derivative of a composite of two functions can be written in the form

In this equation, both and are functions of one variable. Now suppose that is a function of two variables and is a function of one variable. Or perhaps they are both functions of two variables, or even more. How would we calculate the derivative in these cases? The following theorem gives us the answer for the case of one independent variable.

Chain Rule for One Independent Variable

Suppose that and are differentiable functions of and is a differentiable function of Then is a differentiable function of and

where the ordinary derivatives are evaluated at and the partial derivatives are evaluated at

#### Proof

The proof of this theorem uses the definition of differentiability of a function of two variables. Suppose that f is differentiable at the point where and for a fixed value of We wish to prove that is differentiable at and that (Figure) holds at that point as well.

Since is differentiable at we know that

where We then subtract from both sides of this equation:

Next, we divide both sides by

Then we take the limit as approaches

The left-hand side of this equation is equal to which leads to

The last term can be rewritten as

As approaches approaches so we can rewrite the last product as

Since the first limit is equal to zero, we need only show that the second limit is finite:

Since and are both differentiable functions of both limits inside the last radical exist. Therefore, this value is finite. This proves the chain rule at the rest of the theorem follows from the assumption that all functions are differentiable over their entire domains.

Closer examination of (Figure) reveals an interesting pattern. The first term in the equation is and the second term is Recall that when multiplying fractions, cancelation can be used. If we treat these derivatives as fractions, then each product “simplifies” to something resembling The variables that disappear in this simplification are often called intermediate variables: they are independent variables for the function but are dependent variables for the variable Two terms appear on the right-hand side of the formula, and is a function of two variables. This pattern works with functions of more than two variables as well, as we see later in this section.

Using the Chain Rule

Calculate for each of the following functions:

1. To use the chain rule, we need four quantities— and

Now, we substitute each of these into (Figure):

This answer has three variables in it. To reduce it to one variable, use the fact that We obtain

This derivative can also be calculated by first substituting and into then differentiating with respect to

Then

which is the same solution. However, it may not always be this easy to differentiate in this form.

2. To use the chain rule, we again need four quantities— and

We substitute each of these into (Figure):

To reduce this to one variable, we use the fact that and Therefore,

To eliminate negative exponents, we multiply the top by and the bottom by

Again, this derivative can also be calculated by first substituting and into then differentiating with respect to

Then

This is the same solution.

Calculate given the following functions. Express the final answer in terms of

Hint

Calculate and then use (Figure).

It is often useful to create a visual representation of (Figure) for the chain rule. This is called a tree diagram for the chain rule for functions of one variable and it provides a way to remember the formula ((Figure)). This diagram can be expanded for functions of more than one variable, as we shall see very shortly.

Tree diagram for the case

In this diagram, the leftmost corner corresponds to Since has two independent variables, there are two lines coming from this corner. The upper branch corresponds to the variable and the lower branch corresponds to the variable Since each of these variables is then dependent on one variable one branch then comes from and one branch comes from Last, each of the branches on the far right has a label that represents the path traveled to reach that branch. The top branch is reached by following the branch, then the branch; therefore, it is labeled The bottom branch is similar: first the branch, then the branch. This branch is labeled To get the formula for add all the terms that appear on the rightmost side of the diagram. This gives us (Figure).

In (Figure), is a function of and both and are functions of the independent variables

Chain Rule for Two Independent Variables

Suppose and are differentiable functions of and and is a differentiable function of Then, is a differentiable function of and

and

We can draw a tree diagram for each of these formulas as well as follows.

Tree diagram for and

To derive the formula for start from the left side of the diagram, then follow only the branches that end with and add the terms that appear at the end of those branches. For the formula for follow only the branches that end with and add the terms that appear at the end of those branches.

There is an important difference between these two chain rule theorems. In (Figure), the left-hand side of the formula for the derivative is not a partial derivative, but in (Figure) it is. The reason is that, in (Figure), is ultimately a function of alone, whereas in (Figure), is a function of both

Using the Chain Rule for Two Variables

Calculate and using the following functions:

To implement the chain rule for two variables, we need six partial derivatives— and

To find we use (Figure):

Next, we substitute and

To find we use (Figure):

Then we substitute and

Calculate and given the following functions:

Hint

Calculate and then use (Figure) and (Figure).

### The Generalized Chain Rule

Now that we’ve see how to extend the original chain rule to functions of two variables, it is natural to ask: Can we extend the rule to more than two variables? The answer is yes, as the generalized chain rule states.

Generalized Chain Rule

Let be a differentiable function of independent variables, and for each let be a differentiable function of independent variables. Then

for any

In the next example we calculate the derivative of a function of three independent variables in which each of the three variables is dependent on two other variables.

Using the Generalized Chain Rule

Calculate and using the following functions:

The formulas for and are

Therefore, there are nine different partial derivatives that need to be calculated and substituted. We need to calculate each of them:

Now, we substitute each of them into the first formula to calculate

then substitute and into this equation:

Next, we calculate

then we substitute and into this equation:

Calculate and given the following functions:

Hint

Calculate nine partial derivatives, then use the same formulas from (Figure).

Drawing a Tree Diagram

Create a tree diagram for the case when

and write out the formulas for the three partial derivatives of

Starting from the left, the function has three independent variables: Therefore, three branches must be emanating from the first node. Each of these three branches also has three branches, for each of the variables

Tree diagram for a function of three variables, each of which is a function of three independent variables.

The three formulas are

Create a tree diagram for the case when

and write out the formulas for the three partial derivatives of

Hint

Determine the number of branches that emanate from each node in the tree.

### Implicit Differentiation

Recall from Implicit Differentiation that implicit differentiation provides a method for finding when is defined implicitly as a function of The method involves differentiating both sides of the equation defining the function with respect to then solving for Partial derivatives provide an alternative to this method.

Consider the ellipse defined by the equation as follows.

Graph of the ellipse defined by

This equation implicitly defines as a function of As such, we can find the derivative using the method of implicit differentiation:

We can also define a function by using the left-hand side of the equation defining the ellipse. Then The ellipse can then be described by the equation Using this function and the following theorem gives us an alternative approach to calculating

Implicit Differentiation of a Function of Two or More Variables

Suppose the function defines implicitly as a function of via the equation Then

provided

If the equation defines implicitly as a differentiable function of then

as long as

(Figure) is a direct consequence of (Figure). In particular, if we assume that is defined implicitly as a function of via the equation we can apply the chain rule to find

Solving this equation for gives (Figure). (Figure) can be derived in a similar fashion.

Let’s now return to the problem that we started before the previous theorem. Using (Figure) and the function we obtain

Then (Figure) gives

which is the same result obtained by the earlier use of implicit differentiation.

Implicit Differentiation by Partial Derivatives
1. Calculate if is defined implicitly as a function of via the equation What is the equation of the tangent line to the graph of this curve at point
2. Calculate and given
1. Set then calculate and
The derivative is given by

The slope of the tangent line at point is given by

To find the equation of the tangent line, we use the point-slope form ((Figure)):

Graph of the rotated ellipse defined by

2. We have Therefore,

Using (Figure),

Find if is defined implicitly as a function of by the equation What is the equation of the tangent line to the graph of this curve at point

Equation of the tangent line:

Hint

Calculate and then use (Figure).

### Key Concepts

• The chain rule for functions of more than one variable involves the partial derivatives with respect to all the independent variables.
• Tree diagrams are useful for deriving formulas for the chain rule for functions of more than one variable, where each independent variable also depends on other variables.

### Key Equations

• Chain rule, one independent variable
• Chain rule, two independent variables

• Generalized chain rule

For the following exercises, use the information provided to solve the problem.

Let where and Find

Let where and Find and

If and find and

If and find

If and find and express the answer in terms of and

Suppose and where and Find

For the following exercises, find using the chain rule and direct substitution.

Let and Express as a function of and find directly. Then, find using the chain rule.

in both cases

Let where and Find

Let where and Find when and

For the following exercises, find using partial derivatives.

Find using the chain rule where and

Let and Find

Let and Find

Find by the chain rule where and

Let and Find and

and

Let where and Find and

If and find and when and

Find if and

If and find

For the following exercises, use this information: A function is said to be homogeneous of degree if For all homogeneous functions of degree the following equation is true: Show that the given function is homogeneous and verify that

The volume of a right circular cylinder is given by where is the radius of the cylinder and y is the cylinder height. Suppose and are functions of given by and so that are both increasing with time. How fast is the volume increasing when and

The pressure of a gas is related to the volume and temperature by the formula where temperature is expressed in kelvins. Express the pressure of the gas as a function of both and Find when cm3/min, K/min, cm3, and

The radius of a right circular cone is increasing at cm/min whereas the height of the cone is decreasing at cm/min. Find the rate of change of the volume of the cone when the radius is cm and the height is cm.

The volume of a frustum of a cone is given by the formula where is the radius of the smaller circle, is the radius of the larger circle, and is the height of the frustum (see figure). Find the rate of change of the volume of this frustum when

A closed box is in the shape of a rectangular solid with dimensions (Dimensions are in inches.) Suppose each dimension is changing at the rate of in./min. Find the rate of change of the total surface area of the box when

The total resistance in a circuit that has three individual resistances represented by and is given by the formula Suppose at a given time the resistance is the y resistance is and the resistance is Also, suppose the resistance is changing at a rate of the resistance is changing at the rate of and the resistance has no change. Find the rate of change of the total resistance in this circuit at this time.

The temperature at a point is and is measured using the Celsius scale. A fly crawls so that its position after seconds is given by and where are measured in centimeters. The temperature function satisfies and How fast is the temperature increasing on the fly’s path after sec?

The components of a fluid moving in two dimensions are given by the following functions: and The speed of the fluid at the point is Find and using the chain rule.

Let where Use a tree diagram and the chain rule to find an expression for

### Glossary

generalized chain rule
the chain rule extended to functions of more than one independent variable, in which each independent variable may depend on one or more other variables
intermediate variable
given a composition of functions (e.g., the intermediate variables are the variables that are independent in the outer function but dependent on other variables as well; in the function the variables are examples of intermediate variables
tree diagram
illustrates and derives formulas for the generalized chain rule, in which each independent variable is accounted for