Rates Of Change And Small Increments

Differentiation measures how one quantity changes as another changes. But often two or more quantities change together over time, and we want to link their rates: if a balloon's radius grows, how fast does its volume grow? Connected rates of change answer exactly these questions using the chain rule.

In this lesson you will use the chain rule to connect rates, with \(\frac{dy}{dt}=\frac{dy}{dx}\times\frac{dx}{dt}\), and learn how a derivative estimates a small change through \(\delta y \approx \frac{dy}{dx}\,\delta x\). Together they show how calculus handles change that ripples from one quantity to another.