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 $t$ 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:

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

Categories: Calculus | Differential equations

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07-14-2008 23:18:10
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