The Chain Rule is one of the most important tools students will need to evaluate derivatives. The Chain Rule is the basis for implicit differentiation and related rates that students will see in Unit 3 and 4 and it shows up all over the AP Calculus Exam. But the chain rule is still notoriously hard for students and errors abound.
Why is the Chain Rule hard for students?
We'll use an example problem to study the four most common mistakes when using the chain rule and then talk about how to avoid them.
We think these mistakes stem from two main sources. First, students generally have weak conceptual understanding of composite functions. They have not yet internalized what it means that the output of the first (inside) function becomes the input of the second (outside) function (Common Mistake #2). The solution involves changing how we teach composite functions in earlier grades. Check out our approach to teaching composite functions in Algebra 2 in "The Pumpkin Pi Bakery (Part 2)"and in Precalculus in “How Much Does It Cost to Tile a Pool?”.
The second source of confusion is that whatever leverage we’ve gained in the earlier grades related to composite functions, we undo in our typical approach to teaching the chain rule. Most textbooks state the chain rule as follows:
But note how this is different from evaluating composite functions. When we evaluate f(g(3)), we work from the inside out. We input 3 into the function g(x) and then take that output, g(3), and substitute it as the input for f(x). Why do we reverse this for the chain rule by working from the outside in?
Is there a better way?
In our class we teach the chain rule differently. We say that:
The difference may seem negligible. After all, we know that the commutative property works. But this small shift has greatly increased our students’ success with the chain rule. Here’s why:
Since students start by taking the derivative of g(x), they are not likely to forget to take its derivative (Common Mistake #3). Once students take the derivative of the inside function, g(x), they will likely not “reuse” that derivative inside f’ (Common Mistake #1 and #2).
Students move from the inside to the outside, just like they do when evaluating the function! They know to stop taking derivatives when there’s no function left (Common Mistake #4)! This approach also works especially well when there are multiple chains in the function. Students can just keep working their way from the inside to the outside and consider what goes into each subsequent function.
A lesson for teaching the Chain Rule
Our first lesson in Unit 3 asks students: “How is Lindt Chocolate Made?” The analogy of producing Lindt Chocolates helps drive the question of the day: Why can’t we just take the derivative of the outside and inside functions, each with respect to x? As with all composite functions, the output of the first step (inside function) becomes the input of the next step (outside function). This is illustrated by the Lindt production process. The cocoa beans get crushed. The crushed beans get mixed with other ingredients to form a paste. The paste gets tempered to form the chocolate and the chocolate is packaged to create the Lindt. The raw ingredients do not go into the Lindt wrapper! The output of the previous step becomes the input of the next step.
A note about notation:
As you will see on the answer key, we find it helpful to introduce the idea of a box to represent the inside function. The box looks like a single item, so students can more easily recognize it as the independent variable, but the dark box may also conceal some things inside it. Think of the box as being the actual chocolate of the Lindt chocolate. It is the input of the wrapper (the outer function), but it represents more than just the raw material (x), it represents the output of all the previous stages. When we think about the rate of change of the final step in the Lindt Chocolate production process, the wrapping, that change takes effect on the chocolate praline, not the raw cocoa beans. Thus, when students take the derivative of the outer function, they know that it is a derivative with respect to its input, which is the box, or the inside function.
Once students have gone through the EFFL lesson, they are ready for the Chain Rule scavenger hunt on the next day.