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July 1st, 2017, 06:44 AM   #1
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Math Focus: Calculus and Physics
Chain rule concept question

Hello all,

I have a quick question on the chain rule for derivatives. I have used it hundreds of times through out my study of calculus, but embarrassingly I get a little confused when, for example, you're doing a related rates problem and you take the derivative of a function you developed and then you multiply by a derivative after everything is said and done. Like:

V = (1/3) pi r^3

dV/dt = (1/3)pi 3r^2 (dr/dt)

I 100% understand that the when using the chain rule you take the derivative of the "outside" function, and then multiply by the derivative of the "inside" function. But when I am doing problems like this, I fail to see a composition of functions. And then I do not see the reason to utilize the chain rule.

I would be forever thankful if someone could explain why this is the case.

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Jacob
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July 1st, 2017, 08:23 AM   #2
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Both $V$ and $r$ are implicit functions of time.
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July 1st, 2017, 03:06 PM   #3
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The easiest explanation is as follows: since $r$ depends on $t$, we have $r(t)$. The function $V(r)$ is then differentiated respect to time as:
\begin{eqnarray}
V(r(t))\to \frac{dV}{dt}=\frac{dV}{dr}\frac{dr}{dt}
\end{eqnarray}
and that is it. Just remember, for example, how to differentiate
\begin{eqnarray}
\sin(\log(x))
\end{eqnarray}
we argue that $z=\log(x)$, so that the function $\sin(z)$ depends implicitly on the variable $x$ (see comment above). Then we all need to know is the formula I wrote above to differentiate, hope it helps now!
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Last edited by skipjack; July 2nd, 2017 at 03:28 AM.
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