Related Rates

Key idea

Link rates of change of connected quantities.

In depth

When several quantities change with time and are linked by an equation, differentiate with respect to t and substitute known values. Steps: draw, write the relation, differentiate, substitute only after differentiating.

Worked example

A circle’s radius grows at 2 cm/s. A = πr², dA/dt = 2πr dr/dt. At r = 5: 20π cm²/s.

Real-world uses

• Finding how fast water level rises in a cone-shaped tank
• Tracking a ladder sliding down a wall

Common mistake

Plugging in numbers before differentiating.

How to learn this

1. Read the key idea and say it in your own words.
2. Work through the worked example, then try a similar problem.
3. Use the interactive lesson’s visual and mini-quiz to check the pattern.

Frequently asked questions

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Related expert topics in Calculus

• Derivatives of Inverse Trig – Derivatives of arcsin, arccos, arctan.
• Implicit Differentiation – Differentiate both sides of F(x,y)=0 treating y as a function of x.
• Logarithmic Differentiation – Take logs before differentiating products and powers.
• Linear Approximation – Approximate f near a using its tangent line.
• Differentials – Use dy = f′(x)dx for small changes.
• Second Derivative Test – f″ > 0 means concave up; f″ < 0 concave down.