Related rates

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} \times {dx \over dt}.<math>