Related Rates of Change

Starting point: what you already know

• A rate of change is a derivative. $\frac{dr}{dt}$ means "how fast $r$ changes each second".
• The chain rule: $\frac{dy}{dt}=\frac{dy}{dx}\times \frac{dx}{dt}$. Think of the $dx$ parts "cancelling".
• Formulas for shapes: circle area $\pi r^2$, sphere volume $\frac{4}{3}\pi r^3$, sphere surface area $4\pi r^2$, cone volume $\frac{1}{3}\pi r^{2}h$, cube volume $x^3$.
• Similar triangles: matching sides are in the same ratio.
• Signs: a quantity that is getting smaller has a negative rate.

The method

Turning the chain the other way

Often you know how fast the "big" quantity changes and want the small one. Flip the derivative.

Quick summary

• Linked quantities have linked rates.
• Chain rule: $\frac{dy}{dt}=\frac{dy}{dx}\times \frac{dx}{dt}$, and $\frac{dx}{dy}=1÷\frac{dy}{dx}$.
• Find a formula with only the two quantities you need. Use similar triangles or Pythagoras to remove a third.
• Differentiate first, substitute last. Check the units and the sign.

Quiz