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Applications of Derivatives · Theorems

Related Rates

ddtF(x,y)=Fxdxdt+Fydydt\frac{d}{dt}F(x,y) = F_x\frac{dx}{dt} + F_y\frac{dy}{dt}

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.

  • 2 worked examples
  • 15 in Applications of Derivatives

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

1 more worked examplePremium

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?

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Common questions

What is the Related Rates?

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. It is one of the applications of derivatives formulas in the CalcRef reference.

How do you use the Related Rates?

Worked example. A circle has radius growing at 2 cm/s. Find dA/dt when r = 5 cm. A = πr². Differentiate: dA/dt = 2πr · dr/dt. dA/dt = 2π(5)(2) = 20π. Answer: dA/dt = 20π ≈ 62.83 cm²/s.

Practice

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