Calculus · Step-by-step guide

Related Rates

Solve related-rates problems by connecting changing quantities, differentiating with respect to time, and tracking units.

By Tyler Blovat··9 min read
The short answer

Start with the central idea

Draw and label the situation, write one equation connecting the variables, differentiate with respect to time, substitute the values for the instant described, and solve for the requested rate.

01

How it works

Build the method from meaning before memorizing the moves.

These are the decisions that make the procedure reliable. Read them once, then look for each idea in the worked examples below.

1

Variables change with time

Lengths, areas, and volumes are functions of t even when t is not written explicitly.

2

Differentiate before substituting

Substituting changing quantities too early can turn them into constants and erase their rates.

3

Use signed rates

Increasing quantities have positive derivatives; decreasing quantities have negative derivatives.

02

See the structure

A picture makes the relationships easier to remember.

tangent line
The derivative at the marked point is the slope of the highlighted tangent line.
03

Worked examples

Follow the reason for each line, then try to reproduce it without looking.

Example 1Expanding circle

Radius grows at 2 cm/s. Find dA/dt when r = 5.

  1. A=πr2\displaystyle A=\pi r^2.
  2. Differentiate: dA/dt=2πr dr/dt\displaystyle dA/dt=2\pi r\,dr/dt.
  3. Substitute r = 5 and dr/dt = 2.
Answer20π\displaystyle 20\pi cm²/s
Example 2Rising ladder

A 10-ft ladder’s base slides out at 1 ft/s. Find dy/dt when x = 6.

  1. x2+y2=100\displaystyle x^2+y^2=100, so y = 8.
  2. Differentiate: 2xx′+2yy′=0\displaystyle 2x x'+2y y'=0.
  3. Substitute and solve: 12+16y′=0\displaystyle 12+16y'=0.
Answer−3/4\displaystyle -3/4 ft/s
Example 3Sphere volume

Radius increases 0.5 cm/s at r = 4.

  1. V=4πr3/3\displaystyle V=4\pi r^3/3.
  2. dV/dt=4πr2dr/dt\displaystyle dV/dt=4\pi r^2dr/dt.
  3. Substitute the instant values.
Answer32π\displaystyle 32\pi cm³/s
04

Common mistakes—and how to avoid them

Accuracy improves fastest when you know what to check.

  • Plugging in fixed measurements before differentiating.

  • Dropping derivative factors such as dr/dt.

  • Reporting a decreasing rate as positive.

05

Turn the explanation into a skill

Reading creates recognition. Independent practice creates recall.

Free printable worksheet · Answer key included

Related Rates

Practice diagrams, time derivatives, substitution, and rate units.

Preview the worksheet

A strong practice loop: solve one example with the guide open, solve a similar problem without it, explain the method aloud, and return the next day for a short mixed review.

06

Practice problems with worked solutions

Solve each problem on paper, use the hint only if needed, then compare every step.

1

A circle’s radius grows at 2 cm/s. Find dA/dt when r = 5 cm.

Show hint

Differentiate A=πr2\displaystyle A=\pi r^2 with respect to time.

Show worked solution
  1. dA/dt=2πr dr/dt\displaystyle dA/dt=2\pi r\,dr/dt.
  2. Substitute r = 5 and dr/dt=2\displaystyle dr/dt=2.

Answer: 20π\displaystyle 20\pi cm²/s.

2

A sphere’s radius grows at 1 cm/s. Find dV/dt when r = 3 cm.

Show hint

Use V=4πr3/3\displaystyle V=4\pi r^3/3.

Show worked solution
  1. dV/dt=4πr2dr/dt\displaystyle dV/dt=4\pi r^2dr/dt.
  2. Substitute 3 and 1.

Answer: 36π\displaystyle 36\pi cm³/s.

3

A 10-ft ladder slides down a wall. When x = 6 ft and dx/dt = 2 ft/s, find dy/dt.

Show hint

Differentiate x2+y2=100\displaystyle x^2+y^2=100.

Show worked solution
  1. At x = 6, y=8\displaystyle y=8.
  2. 2xx′+2yy′=0\displaystyle 2x x'+2y y'=0.
  3. y′=−(6)(2)/8=−3/2\displaystyle y'=-(6)(2)/8=-3/2.

Answer: −1.5\displaystyle -1.5 ft/s.

4

Water fills a cylinder of radius 4 ft at 8 ft³/min. Find dh/dt.

Show hint

Radius is constant.

Show worked solution
  1. V=π(4)2h=16πh\displaystyle V=\pi(4)^2h=16\pi h.
  2. dV/dt=16πdh/dt\displaystyle dV/dt=16\pi dh/dt.
  3. 8=16πdh/dt\displaystyle 8=16\pi dh/dt.

Answer: 1/(2π)\displaystyle 1/(2\pi) ft/min.

5

A square’s side grows at 3 cm/s. Find dA/dt when s = 10 cm.

Show hint

Differentiate A=s2\displaystyle A=s^2.

Show worked solution
  1. dA/dt=2s ds/dt\displaystyle dA/dt=2s\,ds/dt.
  2. Substitute 10 and 3.

Answer: 60 cm²/s.

07

Frequently asked questions

Quick answers before you move on.

What should I remember first?

Draw and label the situation, write one equation connecting the variables, differentiate with respect to time, substitute the values for the instant described, and solve for the requested rate.

How do I know whether I understand this topic?

You should be able to name the method, explain why each step is valid, complete a new example without copying, and check whether your answer is reasonable.

What should I do if I keep making the same mistake?

Write the error as a specific checkpoint—for example, “I will identify the hypotenuse before substituting.” Then use that checkpoint on three short problems rather than repeating a full page without feedback.

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