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Handout Tuesday 10/7
Let’s think a bit about why the chain rule makes sense. Remember that the chain rule describes how to take the derivative of a composition of functions: \(f(g(x))\text{.}\) For this to make sense, it must be that the output of \(g(x)\) is a reasonable input to \(f(x)\text{.}\) What functions would this make sense for?
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Suppose you own a company that makes artisanal soap. The number of bars of soap you can make is a function of the amount of money you spend on supplies, say \(g(x) = 3x - 20\text{.}\) The amount of money you earn from your soap business is a function of the number of bars of soap you make (and sell): \(f(x) = 7x\text{.}\) What happens to your income if you increase your investment by a dollar when you are already spending $100?
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Another application: the daily water consumption in Greeley is a function of the population: \(w(p) = 30p + 10,000\text{.}\) The population of Greeley is a function of time: \(P(t) = 100,000 \cdot 1.2^t\text{.}\) At what rate is the water consumption increasing as a function of time?
We can also use the limit definition of the derivative to understand the chain rule.
\begin{align*}
\frac{d}{dx}f(g(x)) \amp = \lim_{h\to 0}\frac{f(g(x+h)) - f(g(x))}{h}\\
\amp = \lim_{h\to 0}\frac{f(g(x+h)) - f(g(x))}{g(x+h)-g(x)}\frac{g(x+h)-g(x)}{h}
\end{align*}
Anyway, let’s do some more examples. First a straight forward example, such as \(f(x) = (\sin(x) + x^2)^8\text{.}\) Then a few where you need to use the chain rule together with other rules, like the product rule, the quotient rule, or the chain rule itself.
