Here we work abstract related rates problems.

Suppose we have two, variables and , which are both changing with respect to time. A related rates problem is a problem where we know one rate, say , at a given instant, and wish to find the other (the unknown rate is ”related” to the known rate).

Here, the chain rule is key: If is written in terms of , and we are given , then it is easy to find using the chain rule: In many cases, particularly the interesting ones, our functions will be related in some other way. Nevertheless, in each case we’ll use the chain rule to help us find the desired rate. In this section, we will work several abstract examples in order to emphasize the mathematical concepts involved. In each of the examples below, we will follow essentially the same plan of attack:

Introduce variables, identify the given rate and the unknown rate.

Assign a variable to each quantity that changes in time.

Draw a picture.
If possible, draw a schematic picture with all the relevant information.
Find equations.
Write equations that relate all relevant variables.
Differentiate with respect to t.
Here we will often use implicit differentiation and obtain an equation that relates the given rate and the unknown rate.
Evaluate and solve.
Evaluate each quantity at the relevant instant and solve for the unknown rate.


In order to relate several variables we can use known formulas.

In our next example we consider an expanding circle and use the formulas for perimeter and area of a circle.

Right triangles

In our next example, we consider an expanding right triangle and use the Pythagorean Theorem to relate relevant variables.

Angular rates

We can also investigate problems involving angular rates.

Similar triangles

Finally, facts about similar triangles are often useful when solving related rates problems.