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The radius $r$ and area $A = \pi r^2$ of a circle are both differentiable functions of time $t$. Write an equation relating $\frac{dA}{dt}$ to $\frac{dr}{dt}$.

1. $\frac{dA}{dt} = 2r\,\frac{dr}{dt}$
2. $\frac{dA}{dt} = 2\pi r\,\frac{dr}{dt}$
3. $\frac{dA}{dt} = 4\pi r\,\frac{dr}{dt}$
4. $\frac{dA}{dt} = 3\pi r\,\frac{dr}{dt}$

Correct answer: $\frac{dA}{dt} = 2\pi r\,\frac{dr}{dt}$

Solution

Since $A = \pi r^2$ and $r$ depends on time, differentiate using the chain rule: $\frac{dA}{dt} = \pi \cdot 2r \cdot \frac{dr}{dt}$. This gives $\frac{dA}{dt} = 2\pi r\,\frac{dr}{dt}$.

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