Related rates

Topics in calculus

Fundamental theorem | Function | Limits of functions | Continuity | Calculus with polynomials | 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 | Solids of revolution | 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 t and are related by the equation

y = x^3 + 5,\,

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:

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

Next, apply the chain rule:

{dy \over dt} = 3x^2{dx \over dt}.

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

{dy \over dt} = 3(9)(2) = 54.

Basically,

{dy \over dt} = {dy \over dx} {dx \over dt}.

See also: Related rates, Calculus, Calculus with polynomials, Chain rule, Continuous function, Derivative, Differential calculus, Disk integration, Function (mathematics)