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Pure · Differentiation

Chapter 1 · 4

The idea

Connected rates of change

Linking the rate of change of one quantity to another through a shared variable using the chain rule — dV/dt = (dV/dr)(dr/dt) — for problems like an expanding balloon, a draining cone or a sliding ladder.

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Pure · Differentiation

Connected rates of change

Linking the rate of change of one quantity to another through a shared variable using the chain rule — dV/dt = (dV/dr)(dr/dt) — for problems like an expanding balloon, a draining cone or a sliding ladder.

Why it works

Changing together

Many real situations have several quantities changing together over time: as a balloon inflates, its radius, surface area and volume all grow at once, locked together by geometry. You are usually told one rate and asked for another. The chain rule connects them through the variable they share.

The balloon

Suppose a spherical balloon is inflated so that air is pumped in at dVdt=50 cm3s−1\frac{dV}{dt} = 50\ \text{cm}^3\text{s}^{-1}, and you want how fast the radius grows, drdt\frac{dr}{dt}, when r=5r = 5. Volume and radius are linked by V=43πr3V = \frac43\pi r^3, so dVdr=4πr2\frac{dV}{dr} = 4\pi r^2. The chain rule strings the rates together:

dVdt=dVdr×drdt\frac{dV}{dt} = \frac{dV}{dr} \times \frac{dr}{dt}

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