Connected rates of change
Objective. Relate changing quantities through differentiated constraints.
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- 2. Worked example
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Learn: the key idea
Connected rates of change link two or more variables that change with time through a shared equation, using the chain rule to relate their rates. If a quantity A depends on a quantity r, and both change with time t, then dA/dt = (dA/dr) × (dr/dt). For a sphere with volume V = (4/3)πr³, differentiating gives dV/dr = 4πr², so dV/dt = 4πr² × (dr/dt). Always write the connecting equation first, differentiate it, then substitute known rates.
Worked example
A circle's radius increases at 3 cm/s. Find the rate of change of area when r = 4 cm, using connected rates of change.
- 1Write A = πr² and differentiate: dA/dr = 2πr
- 2State the chain rule link: dA/dt = (dA/dr) × (dr/dt)
- 3Substitute dr/dt = 3 and r = 4: dA/dt = 2π(4)(3)
- 4Simplify to dA/dt = 24π cm²/s
Answer: dA/dt = 24π cm²/s
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Start practisingCommon mistake. Students often substitute numerical values before differentiating, which fixes the variable and gives a derivative of zero. Always differentiate the general equation symbolically first, then substitute specific values afterwards.
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