Related Rates

Differentiate a shared equation in time to link two rates.

The idea

A related rates problem asks for the rate of change of one quantity given the known rate of another, when an equation ties the two quantities together. Differentiating that equation with respect to time converts the known rate into the wanted one.

The conversion is the chain rule. Suppose a disc's area and radius satisfy $A = \pi r^{2}$, and both are functions of time. Differentiating both sides with respect to $t$ treats $r$ as an inner function, so $\frac{dA}{dt} = 2\pi r\,\frac{dr}{dt}.$ The factor $\frac{dr}{dt}$ must be kept: we are differentiating with respect to time, not with respect to $r$, and the radius is itself moving. Dropping it is the standard mistake.

The procedure has three steps. Write an equation relating the quantities. Differentiate it with respect to $t$, attaching a rate to every variable that changes. Only then substitute the values that hold at the instant in question — substituting first freezes a variable that is supposed to be moving, and its rate vanishes from the answer. If the equation carries more changing variables than known rates, the problem's geometry supplies a relation between them; use it to eliminate one variable before differentiating.

Ways to work on it

• Walkthrough. The inflating balloon: link, differentiate in t, substitute.
• Practice. A sliding ladder: find how fast the top moves.
• Hardest. A draining cone: find how fast the water level falls.

Not sure where to start? Take the ten-question placement test.