Products and Chains

Combining rates of change

An interactive lesson with 7 steps, about 6 minutes: you predict, drag and solve, with instant feedback on every answer. It runs in your browser and needs JavaScript.

What you’ll learn

• Product rule: $(fg)' = f'g + fg'$ (the corner piece vanishes in the limit).
• Chain rule: $(f(g(x)))' = f'(g(x))\,g'(x)$: rates multiply.
• Related rates: differentiate an equation linking quantities to link their rates.

Steps

1. A tempting mistake
2. A growing rectangle
3. Use the product rule
4. The chain rule
5. A power of a bracket
6. The sliding ladder
7. Ladder speed