Connected Rates of Change

This lesson covers connected rates of change — using the chain rule to relate rates of change of different variables — as required by the Edexcel A-Level Mathematics specification (9MA0). These problems are among the most challenging at A-Level and require careful application of the chain rule.

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The Core Idea

If two variables are related, their rates of change are connected through the chain rule:

dA/dt = (dA/dr) × (dr/dt)

More generally, if we know the rate of change of one variable and the relationship between variables, we can find the rate of change of another.

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The Chain Rule for Connected Rates

Given a chain of relationships:

Want to find	Know	Chain Rule
dV/dt	dV/dr and dr/dt	dV/dt = (dV/dr) × (dr/dt)
dA/dt	dA/dr and dr/dt	dA/dt = (dA/dr) × (dr/dt)
dV/dt	dV/dh, dh/dr, dr/dt	dV/dt = (dV/dh) × (dh/dr) × (dr/dt)

The key is to construct a chain that links what you want to what you know.

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Worked Example 1: Expanding Circle