Related Rates

You’ll be able to

Relate the rates of change of quantities linked by an equation
Differentiate a geometric relationship implicitly with respect to time
Substitute given values only after differentiating

Rates tied together by an equation

A related-rates problem has several quantities changing over time that are linked by an equation. Differentiating that equation with respect to time t connects their rates. Because every variable depends on t, each differentiation produces a factor like dx/dt or dr/dt — this is implicit differentiation with t as the hidden variable. The goal is to find one unknown rate from the others.

A reliable procedure

Follow a fixed recipe: (1) draw a picture and name the variables; (2) write an equation relating them (often a geometric formula); (3) differentiate both sides with respect to t; (4) substitute the known values and rates; (5) solve for the unknown rate. Do the substitution last — plugging numbers in before you differentiate freezes variables that are actually changing.

Differentiating a relationship in time

If V = (4/3)πr³, then dV/dt = 4πr² · (dr/dt)

Every variable carries a d/dt. The chain rule links dV/dt to dr/dt through the geometry.

Worked example

Air is pumped into a spherical balloon so that its volume increases at 100 cm³/s. How fast is the radius increasing when r = 5 cm? (V = (4/3)πr³)

1.The relating equation is V = (4/3)πr³, with both V and r functions of t.
2.Differentiate with respect to t: dV/dt = 4πr²·(dr/dt).
3.Substitute the known values dV/dt = 100 and r = 5: 100 = 4π(5)²·(dr/dt) = 100π·(dr/dt).
4.Solve for dr/dt: dr/dt = 100/(100π) = 1/π.

Answer: dr/dt = 1/π ≈ 0.318 cm/s. The radius grows slowly even though volume pours in, because the surface area at r = 5 spreads the incoming air over a large sphere.

Watch out

Never substitute the changing values before differentiating. If you set r = 5 first, r becomes a constant, its derivative vanishes, and the relationship between the rates is destroyed. Differentiate the general equation, then plug in.

Checkpoint

A 13-ft ladder leans against a wall; its base slides away at 2 ft/s. With x horizontal and y vertical, which equation correctly relates the rates?

Checkpoint

The area of a circle grows as its radius increases. If A = πr², what is dA/dt in terms of dr/dt?

On the exam

AP related-rates responses must show the differentiated equation with the d/dt factors before any numbers appear. Readers look for that general relationship; a correct final number with no setup earns little credit.

Answer the 2 checkpoints as you read.