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In Chapter 4 of Tom M. Apostol's book Calculus Volume I Second Edition, he discusses the chain rule and its applications. Everything seems fine until related rates.

Why is it "not necessary to express [a variable] as a function of [time] in order to determine the derivative" with respect to time?

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    $\begingroup$ Can you please give an example? The question is a bit unclear without context. $\endgroup$ – Franklin Pezzuti Dyer Jul 15 '17 at 18:19
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For context, the question Apostol asks is, "(Regarding an expanding balloon) How fast is the radius of the balloon increasing when the radius is $5$ cm?

What Apostol is saying is that even though the radius, $r(t)$, is a function of time, we don't need to explicitly need to express $r(t)$ as a function, e.g. $r(t) = 2t$, in related rates problems. We're only interested in the quantity, $\frac{d}{dt}r(t)\biggr|_{t=5}$.

To solve for this, we only need to relate the two expressions for volume and radius together,

$$ \dfrac{dV}{dt}\biggr|_{t=5} = 4\pi(r(5))^2\frac{d}{dt}r(t)\biggr|_{t=5}$$

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    $\begingroup$ Ah. Hence the term implicit differentiation? $\endgroup$ – Chaz Biroan Jul 15 '17 at 18:48
  • $\begingroup$ @ChazBiroan Exactly! $\endgroup$ – Flowsnake Jul 15 '17 at 18:52

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