4.4 Introduction to Related Rates

Linked quantities

Related rates problems involve quantities linked by an equation, all changing with time. Differentiating the relating equation with respect to t (implicit differentiation) produces an equation linking their rates.

First step

The key setup is finding the equation that relates the variables (from geometry or the situation). Example: A = πr² ⇒ dA/dt = 2πr·(dr/dt).

Key terms

Related rates

Quantities linked by an equation, all changing with time.

Differentiate in t

Apply implicit differentiation with respect to time.

First step

Find the equation relating the variables.

Technique

Implicit differentiation.

Example

A = πr² ⇒ dA/dt = 2πr·(dr/dt).

Try an AP-style question

x and y depend on time t, and x² + y² = 25. Differentiating with respect to t gives…

1. x(dx/dt) + y(dy/dt) = 25
2. 2x(dx/dt) + 2y = 0
3. 2x + 2y = 0
4. 2x(dx/dt) + 2y(dy/dt) = 0