Introduction to Related Rates

What Related Rates Problems Are About

Related rates problems describe a situation in which two or more quantities are changing over time. The quantities are connected by an equation, so when one quantity changes, another quantity must change as well.

The key idea is that you are usually not differentiating with respect to . Instead, every variable depends on time, so you differentiate both sides with respect to .

For example, suppose the radius of a circular balloon is increasing. Its area is related to its radius by:

Because the radius is changing, the area is changing too. Related rates lets us determine how fast the area is changing if we know how fast the radius is changing.

The Fundamental Idea

Suppose two quantities and satisfy an equation:

If both variables change with time, then differentiate both sides with respect to :

This is where the chain rule becomes essential. Because is actually a function of time, differentiating with respect to time gives .

The same idea applies to every variable in a related rates problem. If a variable changes over time, its derivative with respect to time appears in the equation.

A Step-by-Step Method

Identify the quantities that are changing. Assign variables to them and make sure you know what each variable represents.
Write an equation relating the quantities. This is the most important setup step. Use the geometry or physical relationship given in the problem.
Differentiate the entire equation with respect to time. Use the chain rule whenever a variable depends on time.
Substitute the known values. Use the specific instant described in the problem only after differentiating.
Substitute the known rates. Pay close attention to positive and negative signs.
Solve for the requested rate. Include the appropriate units in your final answer.

Worked Examples

Example 1: Growing Circle

The radius of a circle is increasing at . How fast is the area increasing when the radius is ?

Step 1: Define the variables. Let be the radius and be the area.

Step 2: Write the relationship. The area of a circle is .

Step 3: Differentiate with respect to time.

Step 4: Substitute the known values.

Step 5: Solve.

Result: The area is increasing at .

Example 2: Expanding Rectangle

The length of a rectangle is increasing at while its width is increasing at . How fast is the area increasing when the length is and the width is ?

Step 1: Define the variables. Let represent length, represent width, and represent area.

Step 2: Write the relationship.

Step 3: Differentiate with respect to time.

Step 4: Substitute the values.

Result:

Example 3: Sliding Ladder

A 10-foot ladder leans against a wall. The bottom of the ladder is moving away from the wall at . How fast is the top of the ladder moving downward when the bottom is 6 feet from the wall?

Step 1: Define the variables. Let be the distance from the wall to the bottom of the ladder and be the height of the top of the ladder.

Step 2: Write the geometric relationship. The ladder forms a right triangle, so .

Step 3: Differentiate with respect to time.

Step 4: Find the corresponding height. When , .

Step 5: Substitute the rates and solve.

Result: The top of the ladder is moving downward at .

Example 4: Volume of a Sphere

The radius of a spherical tank is increasing at . How fast is the volume increasing when the radius is ?

Step 1: Write the volume equation.

Step 2: Differentiate with respect to time.

Step 3: Substitute the known values.

Result:

Why You Must Differentiate Before Substituting

One of the most important habits in related rates is to differentiate first and substitute second.

Suppose you know that a circle has radius at the instant you care about. It is tempting to substitute that value into the original area equation immediately. But if you do that before differentiating, you can accidentally remove the variable relationship that produces the rate you need.

The correct order is:

Only after obtaining the related-rates equation should you plug in the values from the particular instant.

Common Mistakes

Substituting numbers before differentiating

This is one of the most common errors. Keep the variables in the equation while differentiating. Substitute the numerical values only afterward.

Forgetting the chain rule

When differentiating with respect to time, the derivative is , not simply . Every changing variable brings along its rate of change.

Using the wrong sign

If a quantity is decreasing, its rate should normally be negative. For example, if the height of an object is decreasing, then . Do not automatically make every rate positive.

Confusing a quantity with its rate

The radius is a distance. The rate is a distance per unit time. They are different quantities and should not be substituted interchangeably.

Practice Problems

Question 1

The radius of a circle is increasing at . How fast is the area increasing when the radius is ?

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Question 2

The side length of a square is increasing at . How fast is the area increasing when the side length is ?

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Question 3

A 13-foot ladder leans against a wall. The bottom moves away from the wall at . How fast is the top moving when the bottom is 5 feet from the wall?

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Question 4 (Combined Rates)

A rectangular tank has length and width . The length is increasing at and the width is increasing at . How fast is the area changing?

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