Calculus Derivative Rules

Over the past few days I’ve been rethinking my sequencing of introducing derivative rules for the next time I teach calculus.  The impetus for this was an approach I encountered in a Coursera MOOC in Calculus I’m taking this summer to see how a professor would run a Taylor Series-centered calculus class.

Historically, I’ve introduced my high school calculus classes to the product and quotient rules before turing to the chain rule.  I’m now convinced the chain rule should be first because of how beautifully it sets up the other two.

Why the chain rule should be first

Assuming you know the chain rule, check out these derivations of the product and quotient rules.  For each of these, $g_1$ and $g_2$ can be any differentiable functions of x.

PRODUCT RULE:  Let $P(x)=g_1(x) \cdot g_2(x)$.  Applying a logarithm gives,

$ln(P)=ln \left( g_1 \cdot g_2 \right) = ln(g_1)+ln(g_2)$.

Now differentiate and rearrange.

$\displaystyle \frac{P'}{P} = \frac{g_1'}{g_1}+\frac{g_2'}{g_2}$
$\displaystyle P' = P \cdot \left( \frac{g_1'}{g_1}+\frac{g_2'}{g_2} \right)$
$\displaystyle P' = g_1 \cdot g_2 \cdot \left( \frac{g_1'}{g_1}+\frac{g_2'}{g_2} \right)$
$P' = g_1' \cdot g_2+g_1 \cdot g_2'$

QUOTIENT RULE:  Let $Q(x)=\displaystyle \frac{g_1(x)}{g_2(x)}$.  As before, apply a logarithm, differentiate, and rearrange.

$\displaystyle ln(Q)=ln \left( \frac{g_1}{g_2} \right) = ln(g_1)-ln(g_2)$
$\displaystyle \frac{Q'}{Q} = \frac{g_1'}{g_1}-\frac{g_2'}{g_2}$
$\displaystyle Q' = Q \cdot \left( \frac{g_1'}{g_1}-\frac{g_2'}{g_2} \right)$
$\displaystyle Q' = \frac{g_1}{g_2} \cdot \left( \frac{g_1'}{g_1}-\frac{g_2'}{g_2} \right)$
$\displaystyle Q' = \frac{g_1'}{g_2}-\frac{g_1 \cdot g_2'}{\left( g_2 \right)^2} = \frac{g_1'g_2-g_1g_2'}{\left( g_2 \right)^2}$

The exact same procedure creates both rules. (I should have seen this long ago.)

Proposed sequencing

I’ve always emphasized the Chain Rule as the critical algebra manipulation rule for calculus students, but this approach makes it the only rule required.  That completely fits into my overall teaching philosophy:  learn a limited set of central ideas and use them as often as possible.  With this, I’ll still introduce power, exponential, sine, and cosine derivative rules first, but then I’ll follow with the chain rule.  After that, I think everything else required for high school calculus will be a variation on what is already known.  That’s a lovely bit of simplification.

I need to rethink my course sequencing, but I think it’ll be worth it.

7 responses to “Calculus Derivative Rules”

1. This is fantastic. Couple that with Shawn Cornally’s method of introducing the chain rule via playschool gears and I think students would have a much greater understanding of the conceptual meaning of what they are doing.

• Thanks for the Cornally link, John. I’d missed that one. The more I ponder this, the more I think I’m going to dive directly into the Chain Rule — establish it as the big central idea that it is. I’ve got to play more with the gear idea again.

2. Julia

That is SWEET!!! The Chain Rule trumps all.

3. Joe

Yes, definitely. I really like your explanation of the Chain Rule. If possible, can you include a proof o the Chain Rule? Do you teach that to your students? Maybe it’s too difficult, right?

I really like your math blog and the teaching strategies that you post. Can you recommend some ed tech tools? I found technology to be very useful. For example, I have been using this tool called ClassroomIQ (https://classroom-iq.com) lately, and it really helps me to grade homework and exams more quickly and easily. Nice tool to have. Anyway, great post and awesome math!

• Joe,
I do include a proof of the chain rule in all of my calculus classes, although not one university professors would consider water-tight. I’ll try to make that a ‘blog topic sometime soon. As for tech tools, I use many. Handheld graphers, Desmos, Wolfram Alpha, Geogebra & other dynamic geometry, Airsketch & Dragonbox and other apps on iPads, Schoology, and so on. My primary goal with all technology is that it should be seamless in the classroom and very easy to start using, especially for inexperienced or nervous users. I’m happy to share thoughts or insights on any of these if you like. Thanks for your reply.

4. Joe

Thanks for sharing the tools. How do you use WolframAlpha in your classroom? I can see that it’s a powerful tool, but sometimes it gives me a headache when my students try to use that to do homework instead of do it themselves.

• Wolfram Alpha is my classes’ go-to CAS–when we need an answer quickly and aren’t focused on how to manipulate the algebra to get it. Sometimes I even use it for pattern development after which we try to prove the results. For example, determine a formula for the nth derivative of x*e^x & prove thy claim. As for fear of “unwarranted” use on homework, you should just accept that some of them are doing that anyway; you can’t stop it. So I allow and even encourage it. I tell my students that there is never an excuse for not having a correct answer for each homework problem when they walk back in the room. They may not know why an answer is, but they should know the answer. Class time is for filling in blanks. Now they should be able to reproduce the idea behind WHY it works with or without technology (all of my tests are 1/2 tech and 1/2 paper&pencil), so if they blindly plug&chug, ultimately they are hurting their own development. I encourage frequent and responsible technology use. Part of the fun is not knowing what they’ll come back with.