Solve connected rates of change
Differentiate a relationship between changing quantities with respect to time, then substitute the known value and rate.
Cambridge IGCSE Additional Mathematics 0606, 2028–2030 syllabus, Version 1. Paper 1 is non-calculator; Paper 2 requires a scientific calculator. These original lessons provide self-checked topic practice, not a Cambridge grade or a timed paper. Full-paper assessment and human specialist review are still outstanding.
Before you begin
- Cambridge assumes knowledge of IGCSE Mathematics or an equivalent course. Check the domain before using an inverse or logarithm, and retain exact values where the question asks for them.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
If y depends on x and x depends on t, how are dy/dt and dx/dt connected?
What you will learn
- Differentiate a relationship between two changing variables
- Use the chain rule to connect their rates with time
- Attach consistent units to a related rate
Differentiate both sides with respect to time
If area A depends on radius r and radius depends on time t, then dA/dt = (dA/dr)(dr/dt). The chain rule connects the two rates.
For a circle A = πr², so dA/dt = 2πr dr/dt. This relationship gives the instantaneous rate of area change at the current radius, not a fixed rate independent of r.
Substitute only after differentiating
Keep variables and rates symbolic until the relationship has been differentiated. Then insert the radius and its rate at the same instant.
Check units: if r is metres and dr/dt is metres per second, dA/dt has square metres per second. Convert time units before combining rates from different sources.
Pause and explain
What are the units of dA/dt if A is in m² and t is in seconds?
Worked example
A circular pool has radius 5 m and is expanding at 0.2 m/s. Find the rate at which its area is increasing at that instant.
Show the worked solution
- A = πr², so dA/dt = 2πr dr/dt.
- Substitute r = 5 and dr/dt = 0.2.
- Evaluate 2π × 5 × 0.2 and attach area-rate units.
Answer dA/dt = 2π m²/s.
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
The circumference of a circle is C = 2πr. If dr/dt = 0.4 cm/s, find dC/dt.
Give me a hint
Differentiate C with respect to time and then substitute the radius rate.
Compare my reasoning
- dC/dt = 2π dr/dt.
- Substitute dr/dt = 0.4.
- Evaluate 0.8π and attach length-per-time units.
dC/dt = 0.8π cm/s.
Look for these in your work
- Relationship differentiated
- Rate substituted
- Units retained
A cube has volume V = x³, where x is its edge length. Find dV/dt when x = 3 m and dx/dt = 2 cm/s; convert units first.
Give me a hint
Convert dx/dt to metres per second before differentiating or substituting.
Compare my reasoning
- dx/dt = 0.02 m/s.
- dV/dt = 3x² dx/dt.
- At x = 3, dV/dt = 27 × 0.02 = 0.54 m³/s.
dV/dt = 0.54 m³/s.
Look for these in your work
- Length-rate unit converted
- Chain rule applied
- Volume-rate unit stated
The area of a square is A = s². At the instant s = 8 cm, its area is increasing at 3 cm²/s. Find ds/dt.
Give me a hint
Differentiate A with respect to time, then rearrange for the side-length rate.
Compare my reasoning
- dA/dt = 2s ds/dt.
- 3 = 2(8) ds/dt.
- Divide by 16.
ds/dt = 3/16 cm/s.
Look for these in your work
- Rate equation formed
- Known area rate substituted
- Unknown rate isolated with units
Common mistakes
- Substituting numerical values before differentiating
- Forgetting dr/dt in the chain rule
- Using area units for an area rate
- Treating the rate as constant while the changing variable changes
What can you explain now?
If A = πr², what is dA/dt?
Compare with the explanation
2πr dr/dt
Differentiate with respect to time using the chain rule.
After trying it yourself, choose your next review. This is your self-assessment.
Your review choice appears on Today. Sign in to sync it across devices.
Make it stick
How does the chain rule connect rates of two changing quantities, and when should values be substituted?
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.