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Related rates

When two quantities are linked and both change with time, their rates of change are linked too. Differentiate the relation with respect to time using the chain rule, and insert the known values at the end.

\frac{dy}{dt} = \frac{dy}{dx}\cdot\frac{dx}{dt}

chain rule for time rates

\frac{dA}{dt} = 2\pi r\,\frac{dr}{dt}

example: area of a growing circle

Symbols

$\frac{dx}{dt}$	how fast x changes
$t$	time	s

Example

An oil slick grows with $dr/dt = 0.5$ m/s. When $r = 4$ m:

$dA/dt = 2\pi\cdot 4\cdot 0.5 \approx 12.6$ m²/s.

Insert the numbers only after you have differentiated.

Practise applications of derivatives for free →

← Optimization · Newton's method →

Part of Calculus: Applications of derivatives.