Section Week 7 (10/6 - 10/10)
This week we will learn the chain rule (section 2.5) and then apply it to find rules for a few more functions (section 2.6).
Handout Monday 10/6
Today we learned the chain rule and went through many examples. See section 2.5 of the textbook for such exampels.
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.
Handout Wednesday 10/8
Let’s review the chain rule with a few examples, and then see how it can teach us rules for new functions.
Find the derivatives of the following functions.
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\(\displaystyle f(x) = (\sin(x) + 3^x)^5\)
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\(\displaystyle f(x) = (\sin(x)3^x)^5\)
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\(\displaystyle f(x) = \left(\frac{\sin(x)}{3^x}\right)^5\)
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\(\displaystyle f(x) = \cos(x^2 + 7^{\sqrt[3]{x}})\)
Here is another interesting example: \(f(x) = \frac{1}{\cos(x)}\text{.}\) We have done this already using the quotient rule (to find the derivative of \(\sec(x)\)), but notice we could also write this as \(f(x) = (\cos(x))^{-1}\) and then it is a chain rule problem!
Now for something new. Use the chain rule to find the derivative of \(e^{\ln(x)}\text{.}\) Yeah, I know. We don’t the derivative of \(\ln(x)\text{.}\) So just leave that as \([\ln(x)]'\text{.}\)
But wait! \(e^{\ln(x)} = x\) (why?). So this derivative will be the same as \((x)' = 1\text{.}\) So now solve for \([\ln(x)]'\)
Handout Friday 10/10
Today in class we will finish section 2.6 to learn the derivative rules for \(\arctan(x)\) and \(\arcsin(x)\text{.}\)
Handout Derivative Rules
Rules for specific types of functions.
- Constant functions
- \(\displaystyle (k)' = 0\)
- Power functions
- \(\displaystyle (x^{n})' = nx^{n-1}\)
- Exponential functions
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\((a^{x})' = a^{x}\ln(a)\)\((e^{x})' = e^{x}\)
- Logarithmic functions
- \(\displaystyle (\ln(x))' = \frac{1}{x}\)
- Trigonometric functions
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\((\sin(x))' = \cos(x)\text{.}\)\((\cos(x))' = -\sin(x)\text{.}\)\((\tan(x))' = \sec^{2}(x) = \frac{1}{\cos^{2}(x)}\text{.}\)\((\arcsin(x))' = \frac{1}{\sqrt{1-x^{2}}}\text{.}\)\((\arctan(x))' = \frac{1}{1+x^{2}}\text{.}\)
Rules for combinations of functions.
- Constant multiples
- \(\displaystyle (k\cdot f(x))' = k\cdot f'(x)\)
- Sum and difference
- \(\displaystyle (f(x) \pm g(x))' = f'(x) \pm g'(x)\)
- Products (the product rule)
- \(\displaystyle (f(x)\cdot g(x))' = f'(x)g(x) + f(x)g'(x)\)
- Quotients (the quotient rule)
- \(\displaystyle \left(\frac{f(x)}{g(x)}\right)' = \frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^{2}}\)
- Compositions (the chain rule)
- \(\displaystyle (f(g(x)))' = f'(g(x))\cdot g'(x)\)
