There might be light at the end of the Chain Rule tunnel… maybe.

This is going to be a half-formed post. I wanted to get a conceptual way for kids to grok why the chain rule works in calculus. But without doing too much handwaving. And I wanted something visual.

The hard part is: if we have a function $g(f(x))$ , we can approximate the derivative at a particular point by doing the following.

Find two points close to each other, like $(x,g(f(x))$ and $(x+0.001,g(f(x+0.001))$ .

Find the slope between those two points: $\frac{g(f(x+0.001)-g(f(x))}{(x+0.001)-x}$ .

There we go. An approximation for the derivative! (We can use limits to write the exact expression for the derivative if we want.)

But that doesn’t help us understand that $\frac{d}{dx}[g(f(x)]=g'(f(x))f'(x)$ on any level. They seem disconnected!

But I’m on my way there. I’m following things in this way: $x \rightarrow f \rightarrow g$

Check out this thing I whipped up after school today. The diagram on top does $x \rightarrow f$ and the diagram on the bottom does $f \rightarrow g$ . The diagram on the right does both. It shows how two initial inputs (in this case, 3 and 3.001) change as they go through the functions f and g.

At the very bottom, you see the heart of this. It has $\frac{\Delta g}{\Delta f}\cdot\frac{\Delta f}{\Delta x}=\frac{\Delta g}{\Delta x}$
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And then I thought: okay, this is getting me somewhere, but it’s to abstract. So I went more concrete. So I started thinking of something physical. So I went to how maybe someone is heating something up, and in three seconds, the temperature rises dramatically. The temperature measurements are made in Farenheit, but you are a true scientist at heart and want to see how the temperature changed in Celcius.

I love this. I’m proud of this page.

And then of course when I got home, I wanted to see this process visualized, so I hopped on Geogebra and had fun creating this applet (click here or on the image below to go to the applet). These sorts of input-output diagrams going from numberline to numberline are called dynagraphs. You can change the two functions, and you can drag the two initial points on the left around. (The scale of the middle and right bar change automatically with new functions you type! Fancy!)

And of course after doing all this, I remembered watching a video that Jim Fowler made on the chain rule for his online calculus course, and yes, all my thinking is pretty much recapturing his progression.

This, to be clear, is about the fourth idea I’ve had as I’ve been thinking about how to conceptually get at the chain rule for my kids. The other ideas weren’t bad! I just didn’t have time to blog about them, but I also abandoned them because they still felt too tough for my kids. But I think this approach has some promise. It’s definitely not there yet, and I don’t know if I’ll have time to get there this year (so I might have to work on it for next year). But I know to get there I’ll have to focus on making the abstract very tangible, and not have too many logical leaps (so the chain of logic gets lost).

If I’m going to create something I’m proud of, kids are going to have to come out saying “oh, yeah… OBVIOUSLY the chain rule makes sense.” Not “Oh, I guess we did a lot of stuff and it all worked out, so it must be true.”

A blogpost of unformed thoughts, and an applet. Sorry, not sorry. This is my process!