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9.10PureStretch

Connected Rates of Change

When several quantities change with time and are linked by an equation, the chain rule connects their rates. This is the basis of many real-world “how fast is …” problems.

26 min Differentiation
Edexcel A level Maths: 9.10 Connected Rates of ChangeCurated by Kepler, not produced by us. Watch it before the notes below.
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What you'll be able to do

  • Link rates with the chain rule
  • Set up dV/dt = dV/dr · dr/dt type relations
  • Identify the known and required rates
  • Solve connected-rate problems
1

Connecting rates

If quantities are related, their rates of change are connected by the chain rule. For example, dVdt=dVdr⋅drdt\dfrac{dV}{dt} = \dfrac{dV}{dr}\cdot\dfrac{dr}{dt}.

dVdt=dVdr⋅drdt\frac{dV}{dt} = \frac{dV}{dr}\cdot\frac{dr}{dt}
Chain rule linking rates.
Example 1A sphere’s radius grows at 22 cm/s. Find dVdt\dfrac{dV}{dt} when r=3r = 3. (V=43πr3V = \tfrac43\pi r^3.)
1dVdr=4πr2\tfrac{dV}{dr} = 4\pi r^2.
2dVdt=4πr2⋅drdt=4π(9)(2)\tfrac{dV}{dt} = 4\pi r^2 \cdot \tfrac{dr}{dt} = 4\pi(9)(2).
Answer
72π72\pi cm³/s

Tip — Write down each rate you know and the one you want, then bridge them with the chain rule.

Formula recap

dAdt=dAdx⋅dxdt\frac{dA}{dt} = \frac{dA}{dx}\cdot\frac{dx}{dt}
General connected rates.
V=43πr3⇒dVdr=4πr2V = \tfrac43\pi r^3 \Rightarrow \frac{dV}{dr} = 4\pi r^2
Sphere example.

Common mistakes to avoid

Adding the rates instead of using the chain rule.
Multiply the linking derivatives.
Substituting the value of r before differentiating.
Differentiate first, then substitute.

Key takeaways

  • Connected rates link via the chain rule.
  • dV/dt = dV/dr · dr/dt.
  • List known/required rates, then bridge them; substitute last.

Test yourself

Answer a few questions, then review anything you missed.

Prefer paper? Open the written self-check

Answer each question before opening its answer. This self-check does not record a quiz score.

  1. Connected rates of change use the:

    • chain rule
    • product rule
    • quotient rule
    • sine rule
    Show answer and explanation

    Answer: chain rule

    Bridges the rates.

  2. dVdt\frac{dV}{dt} equals:

    • dVdr⋅drdt\frac{dV}{dr}\cdot\frac{dr}{dt}
    • dVdr+drdt\frac{dV}{dr}+\frac{dr}{dt}
    • drdV⋅dtdr\frac{dr}{dV}\cdot\frac{dt}{dr}
    • dVdr\frac{dV}{dr}
    Show answer and explanation

    Answer: dVdr⋅drdt\frac{dV}{dr}\cdot\frac{dr}{dt}

    Chain rule.

  3. For V=43πr3V = \tfrac43\pi r^3, dVdr=\frac{dV}{dr} =

    • 4πr24\pi r^2
    • 43πr2\tfrac43\pi r^2
    • 3πr23\pi r^2
    • 4πr4\pi r
    Show answer and explanation

    Answer: 4πr24\pi r^2

    Differentiate.

  4. With dr/dt = 2 and r = 3, dV/dt =

    • 72π72\pi
    • 36π36\pi
    • 24π24\pi
    • 18π18\pi
    Show answer and explanation

    Answer: 72π72\pi

    4π·9·2.

  5. You should substitute the value of r:

    • after differentiating
    • before differentiating
    • never
    • into dr/dt
    Show answer and explanation

    Answer: after differentiating

    Differentiate first.

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