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Applications of Derivatives · Theorems

Related Rates

ddt[equation] -differentiate both sides with respect to time\frac{d}{dt}[\text{equation}] \text{ -differentiate both sides with respect to time}

Related rates problems involve finding how fast one quantity changes given how fast another changes. Differentiate an equation relating the quantities with respect to time.

Worked examples

A circle has radius growing at 2 cm/s. Find dA/dt when r = 5 cm.
  1. A = πr². Differentiate: dA/dt = 2πr · dr/dt
  2. dA/dt = 2π(5)(2) = 20π

Answer: dA/dt = 20π ≈ 62.83 cm²/s

A 10-ft ladder slides down a wall. The base moves at 1 ft/s. How fast does the top slide when the base is 6 ft from the wall?
  1. x² + y² = 100. When x = 6, y = 8.
  2. Differentiate: 2x(dx/dt) + 2y(dy/dt) = 0
  3. 2(6)(1) + 2(8)(dy/dt) = 0 → dy/dt = -12/16 = -3/4

Answer: The top slides down at 3/4 ft/s.

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