Related rates

Topics in calculus

Fundamental theorem | Function | Limits of functions | Continuity | Mean value theorem | Vector calculus | Tensor calculus

Differentiation

Product rule | Quotient rule | Chain rule | Implicit differentiation | Taylor's theorem | Related rates

Integration

Integration by substitution | Integration by parts | Integration by trigonometric substitution | Integration by disks | Integration by cylindrical shells | Improper integrals | Lists of integrals

In differential calculus, related rates problems involve ratios of derivatives of two or more related variables that are changing with respect to time.

For example, suppose x and y are both differential functions of time <math>t</math> and are related by the equation

<math>y = x^3 + 5,\,</math>

and you are supposed to find dy/dt when x = 3, given that dx/dt = 2 when x = 3.

First, differentiate both sides of the equation with respect to t:

<math>{d \over dt}[y] = {d \over dt}[x^3 + 5].\,</math>

Next, apply the chain rule:

<math>{dy \over dt} = 3x^2{dx \over dt}.</math>

Finally, substitute 3 for x and 2 for dx/dt:

<math>{dy \over dt} = 3(9)(2) = 54.</math>

Basically,

<math>{dy \over dt} = {dy \over dx} {dx \over dt}.</math>