Related Rates

A related-rates problem involves at least two quantities that are changing with time and are linked to each other through some equation; knowing the rate at which some of the quantities change lets you find the rate at which another one changes.

Standard workflow.

1. Write down the equation relating the quantities (a geometric formula — area, volume, Pythagorean relation, similar triangles, …).
2. Differentiate both sides with respect to time $t$, applying the chain rule to every quantity that is itself a function of $t$ (this is the step that actually introduces the rates $\dfrac{d(\cdot)}{dt}$ into the equation).
3. Substitute the values that are known at the instant in question (including any value of the quantity itself, found if necessary from the original constraint equation).
4. Solve algebraically for the required rate.

Worked patterns from the textbook examples, all reused in Exercise 7.1:

• Sphere/circle growth: $V=\tfrac43\pi r^3\Rightarrow \dfrac{dV}{dt}=4\pi r^2\dfrac{dr}{dt}$; $A=\pi r^2\Rightarrow\dfrac{dA}{dt}=2\pi r\dfrac{dr}{dt}$.
• Price vs. supply: differentiate the given price–supply relation with respect to time, then substitute the known $x$ and $\dfrac{dx}{dt}$.
• Conical pile (fixed height-to-radius ratio): use the given ratio to write $V$ in terms of a single variable before differentiating.
• Two-variable related rate (two moving objects): set up a Pythagorean (or similar) relation between the two independently-moving legs and the connecting distance, differentiate implicitly with respect to $t$, then substitute the given instantaneous positions and rates. …