Chain Rule: Experiments, Proof, Practice

Part One: The Plans
Tomorrow I start our third unit in calculus with an introduction to the chain rule. This deep into the semester, I have usually lost most of my creative energy, and don't find time to think this lesson out carefully. I'm teaching great groups of students this semester, which is helping keep my enthusiasm high. So today I'm planning my chain rule lesson.

Describing Composition
First, some functions that we cannot find the derivatives for, because there's more than the variable inside the function. Like y=sin(2x). Here we will review the language of composition.

Experiment by Graphing
Then a graph of the function, to see if we can figure out what the derivative ought to be.

I don't usually write blog posts before I teach the class. This is an interesting point in my process. Where I go next with this may depend somewhat on the students' response to the graph exercise. I know right now that I'm not sure of their response, and that their response may inspire me to move a different direction than what I'm planning right now.

Experiment Algebraically
Where I think I'll go is this. We next take a look at functions which exhibit composition, but don't really have to (ie, they can be simplified). My three examples for now are:

y = (5x-6)²

y = (2x)³

and

y = √(9x)

I'm going to work through finding the derivative without chain rule, and trying to get the derivative in a form that matches the function. I think this will be another way to experimentally figure out the chain rule. 

Prove It

Then we'll do the proof. I found a much nicer proof here (click on the discussion link) than the one in our textbook (Anton). Except that one vital step seems to be missing. 

Practice It

And after that, we'll do lots of practice. The textbook is probably fine for that, if I can't think up enough examples on my feet.

[I found some inspiring posts, but I'm not sure if or when I can use them: Sam's filing cabinet led me to lots of good posts, one of which was this on the monkey and the mathematician. I might suggest my students check this linked wheels applet out on their own.]

Part Two: How It Went
I got up to the first algebraic example on Monday. On Tuesday, we did the three algebraic examples. They weren't enthusiastic about walking through the proof, so I asked them to look it over on the handout shown below. We finished up by practicing with lots of examples.

   Calculus: Chain Rule (for composition of functions)