Mathematics

Problem

Which expression equals

\dfrac{dx}{dt}

for a particle whose position in Cartesian coordinates is given by

x=r\cos\theta

and

y=r\sin\theta

, where

r=r(\theta)

and

\theta=\theta(t)

? a)

-r\sin\theta\,\dfrac{d\theta}{dt}

\left(\dfrac{dr}{d\theta}\cos\theta - r\sin\theta\right)

\left(\dfrac{dr}{d\theta}\cos\theta - r\sin\theta\right)\dfrac{d\theta}{dt}

\left(\dfrac{dr}{d\theta}\cos\theta + r\sin\theta\right)\dfrac{d\theta}{dt}

Intuitive explanation

You're converting a position given in polar form back to Cartesian, then finding how $x$ changes with time. The key is the chain rule: $x$ depends on $t$ through both $r$ and $\theta$ , so you must account for how each one changes. Differentiate $x=r\cos\theta$ with respect to $t$ by treating $r$ and $\theta$ as functions of $t$ , then use the product and chain rules to separate out $\frac{dr}{dt}$ and $\frac{d\theta}{dt}$ .

Intuitive explanation

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