At this point in the term, we have developed essentially all of the major abstract and theoretical tools for differentiation, and have a few little loose ends to tie up before we start looking at applications of the derivative (e.g. applied mathematics problems, curve sketching, and so on).
What I Taught
The first 15 minutes of class were spent on a single problem from the last quiz. I asked them to find the limit
\[
\lim_{\varphi\to-1}\frac{\sin(1+\varphi)}{\varphi^2-1}.
\]
This was meant to be a somewhat difficult question, with a lot of little things going on, but I did give them a hint: define \(x=1+\varphi\). Replace every occurrence of \(\varphi\) with the appropriate expression containing \(x\), then use the identity
\[
\lim_{x\to 0} \frac{\sin(x)}{x} = 1.
\]
I kind of assumed that they would just do the substitution and move on. They didn’t, and so this question turned into a really fruitful discussion of limits, algebra, and trig identities. As I said to my students, it was a hard question, but I don’t regret asking it. (And I was generous with the partial credit, so they really shouldn’t have any reason to complain, anyhow).
We then spent another 20 minutes going over homework problems about implicit differentiation, which was also a productive use of time. Finally, we got into the hodge-podge of topics for the day: derivatives of logarithmic functions, logarithmic differentiation, a generalized power rule, and viewing \(e\) as a limit.
Because I knew that the lecture was a bit scattered, I reserved half of one of my three blackboards for writing down “Fun Facts.” As the lecture progressed, I added things to the list of important ideas that they should know—both to highlight the big ideas from the lecture, and to give them something to study for the next exam.
The first fun fact was the derivative
\[
\frac{d}{dx} \log_{a}(x) = \frac{1}{x\ln(a)},
\]
which we found using implicit differentiation. We had to dig deep into our collective memory to recall that if \(y = \log_a(x)\), then \(a^y = x\), but, having done that, the rest was pretty straightforward. From this, we were also able to generate the derivatives
\[
\frac{d}{dx} \ln(x) = \frac{1}{x},
\qquad
\frac{d}{dx} \ln|x| = \frac{x}{x},
\qquad\text{and}\qquad
\frac{d}{dx} \ln(g(x)) = \frac{g'(x)}{g(x)}
\]
without too much difficulty.
I used these fun facts to work several examples on the board, finishing with the problem
\[
\frac{d}{dx} \ln\left(\frac{x+1}{\sqrt{x-2}}\right).
\]
Initially, we hit this problem with chain rule and the quotient rule, which led to a gnarly bit of algebra to simplify. No one liked that. I asked if there was a better way, and suggested that the mantra I’ve been trying to get them to adopt (“Algebra first, then calculus.”) might be helpful.
Going back to the original problem, we can use properties of logarithms to write
\[
\frac{d}{dx} \ln\left(\frac{x+1}{\sqrt{x-2}}\right)
= \frac{d}{dx} \left[\ln(x+1) – \frac{1}{2}\ln(x-2)\right].
\]
The quotient is now a difference, meaning that we don’t need to use the quotient rule. The derivatives are almost trivial at this point, and the answer comes (nearly) pre-simplified.
This is a useful property of logarithms: products and quotients become sums and differences, and exponents become multiplications. Combining properties of logarithms with the notion of implicit differentiation, we have a way of differentiating really complicated functions as follows: if \(y = f(x)\), then we may find \(y’\) by taking logs, differentiating implicitly, and solving for \(y’\). Such a technique is often easier and less error-prone than the straight-forward approach.
I demonstrated this with a truly nasty derivative, then used the technique to prove the generalized power rule. Previously, we proved the power rule \(\frac{d}{dx} x^n = nx^{n-1}\) for \(n\in\mathbb{Z}_+\), and stated that it isn’t too much more difficult to prove it for \(n\in\mathbb{Q}\), but a generalized power rule for \(n\in\mathbb{R}\) required some more tools. Logarithmic differentiation is the last of the necessary tools, and so here we are.
I finished the lecture by noting that the functions \(a^n\), \(a^{g(x)}\), \([f(x)]^n\), and \([f(x)]^{g(x)}\) are all very different animals, and require slightly different techniques for differentiation. I finished with the example \(\frac{d}{dx} x^{\sqrt{x}}\) by way of demonstration.
I had planned to show them how to derive the limit
\[
e = \lim_{n\to\infty} \left(1+\frac{1}{n}\right)^n
\]
using the techniques discussed above, but didn’t have time for it. Since I had not intended to ask questions about this limit on any exams, I am not hugely disappointed that I didn’t get it (and, in fact, had put it in my notes as a bonus), but it is an interesting result, so I suggested that they look it up themselves, and gave a reference.
What Worked
First off, I am really glad to have spent the time to go over the quiz question. We were able, with one question, to clear up a lot of lingering confusion about limits, and highlight some important algebra concepts that have, it seems, been a problem.
Second, the list of “Fun Facts” was a good idea. It was a source of entertainment at the beginning of the class (yay engagement? maybe?), and ended up being a handy tool to refer back to throughout the class. Oddly enough, it was entirely spontaneous, and was not something that I had planned in advance, so I should try to remember the concept.
Finally, the derivation of
\[
\frac{d}{dx} \ln|x| = \frac{1}{x}
\]
turned out to be far more interesting than I expected. Though we have previously worked with \(|x|\) and its derivative, it seems that another encounter was worthwhile. It was a little frustrating to rederive the derivative of \(|x|\), as we have done it before, but I heard audible clicks this time around, and it was part of a larger derivation, so, at the end of the day, it was time well spent.
What Didn’t Work
I really, really, really should have had them work out examples on their own, rather than simply doing them at the board. I knew it ahead of time, had planned for it, then chickened out at the last minute because I was worried that it would be (a) too time consuming and (b) frustrating for the students, since the concept was new. Neither is a good reason. Let the students do it!