Calculus/Related Rates
Contents
Introduction[edit]
One useful application of derivatives is as an aid in the calculation of related rates. What is a related rate? In each case in the following examples the related rate we are calculating is a derivative with respect to some value. We compute this derivative from a rate at which some other known quantity is changing. Given the rate at which something is changing, we are asked to find the rate at which a value related to the rate we are given is changing.
Process for solving related rates problems:
 Write out any relevant formulas and information.
 Take the derivative of the primary equation with respect to time.
 Solve for the desired variable.
 Plugin known information and simplify.
Notation[edit]
Newton's dot notation is used to show the derivative of a variable with respect to time. That is, if is a quantity that depends on time, then , where represents the time. This notation is a useful abbreviation in situations where time derivatives are often used, as is the case with related rates.
Examples[edit]
Example 1:
 Write out any relevant formulas or pieces of information.
 Take the derivative of the equation above with respect to time. Remember to use the Chain Rule and the Product Rule.
Example 2:
 Write out any relevant formulas and pieces of information.
 Take the derivative of both sides of the volume equation with respect to time.

= =
 Solve for .
 Plugin known information.
Example 3:
Note: Because the vertical distance is downward in nature, the rate of change of y is negative. Similarly, the horizontal distance is decreasing, therefore it is negative (it is getting closer and closer).
The easiest way to describe the horizontal and vertical relationships of the plane's motion is the Pythagorean Theorem.
 Write out any relevant formulas and pieces of information.
 (where s is the distance between the plane and the house)
 Take the derivative of both sides of the distance formula with respect to time.
 Solve for .

=
 Plugin known information

= = = ft/s
Example 4:
 Write down any relevant formulas and information.
Substitute into the volume equation.

= = =
 Take the derivative of the volume equation with respect to time.
 Solve for .
 Plugin known information and simplify.

= = ft/min
Example 5:
 Write out any relevant formulas and information.
Use the Pythagorean Theorem to describe the motion of the ladder.
 (where l is the length of the ladder)
 Take the derivative of the equation with respect to time.
 ( is constant so .)
 Solve for .
 Plugin known information and simplify.

= = ft/sec
Exercises[edit]