I think I now have a way that might help students to get conceptually understand what’s going on. (The gear thing doesn’t help me get it… Although I understand the analogy, it feels divorced from the actual functions themselves… and these functions have a constant rate of change.) But I have never yet found a way to conceptually get them to understand it without confusing them. I have found ways to help kids remember the chain rule (“the outer function is the mama, the inner function is the baby… when you take the derivative, you derive the mama and leave the baby inside, and then you multiply by the derivative of baby”), ways to write things down so their information stays organized, and I have shown them enough patterns to let them see it’s true.

I’m soon going to embark on teaching the chain rule in calculus.