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Related Rates

Objectives

  • Express changing quantities in terms of derivatives.
  • Find relationships among the derivatives in a given problem.
  • Use the chain rule to find the rate of change of one quantity that depends on the rate of change of other quantities.

Summary

When two or more related quantities are changing as implicit functions of time, their rates of change can be related by implicitly differentiating the equation that relates the quantities themselves. For instance, if the sides of a right triangle are all changing as functions of time, say having lengths x , y , and z then these quantities are related by the Pythagorean Theorem: x 2 + y 2 = z 2 . It follows by implicitly differentiating with respect to t that their rates are related by the equation

2 x d x d t + 2 y d y d t = 2 z d z d t ,

so that if we know the values of x , y , and z at a particular time, as well as two of the three rates, we can deduce the value of the third.

See the Desmos Demonstration on Related Rates-Spherical Balloon.

See the Desmos Demonstration on Related Rates-Area & Volume.

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