A Level Maths: Connected Rates of Change Topic Summary and Resources
Video Lessons
Watch alongside the worksheet for the full lesson experience, then test your understanding with the lesson questions.
Revision Notes
Handwritten notes summarising the key ideas for each lesson. Ideal for quick review before a test.
Exam Questions
Past-paper-style questions organised by topic, with full mark schemes.
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Drawn from OCR and Edexcel past papers but designed to be useful for students of all UK exam boards — including AQA and OCR MEI — unless a sheet is explicitly board-specific.
Before You Start This Topic
It will help if you are confident with the following:
- Differentiation — chain rule is the foundation
- Functions — relationships between variables
- Trigonometry — many problems involve angles and trig relationships
A Level Maths connected rates of change is the elegant application of the chain rule to real-world problems where multiple quantities change together. If you know how one quantity changes with time, and how it relates to another quantity, the chain rule tells you how the second changes with time. This appears constantly in physics, engineering, and biology — and is one of the cleanest demonstrations of why calculus is so useful.
The headline technique is the chain rule for rates: if $y$ depends on $x$, and $x$ depends on $t$, then $\frac{dy}{dt} = \frac{dy}{dx} \cdot \frac{dx}{dt}$. For three quantities like volume $V$, radius $r$, and time $t$ — where $V = \tfrac{4}{3}\pi r^3$ — you write $\frac{dV}{dt} = \frac{dV}{dr} \cdot \frac{dr}{dt}$. Given any two of these rates, you find the third. Common contexts include balloons being inflated (volume vs radius vs time), liquids being poured into conical containers, ladders sliding down walls, and shadows being cast by moving objects. The strategy is: write down the relationship between the quantities, differentiate implicitly (or directly), then substitute known values.
Connected rates of change is part of the Pure Maths strand of A Level Maths for AQA, Edexcel, OCR, and OCR MEI students.
Watch out for…
A few things to be careful with: do NOT substitute specific numerical values into the relationship before differentiating — differentiate the general formula first, THEN substitute; signs matter — a tank emptying has $\frac{dV}{dt} < 0$; check the units of your rates (cm/s vs m/s, etc.) and ensure consistency; and pay attention to which rate the question asks for — students often find the wrong one of $\frac{dV}{dr}$ or $\frac{dr}{dt}$.