Calculate derivatives of compositions of differentiable functions
I can identify the inside and outside function of a composition of functions
I can explain why the chain rule is needed to calculate the derivative of a composition of functions.
I can apply the chain rule to find a derivative.
Quick Lesson Plan
This lesson introduces students to the chain rule through the analogy of a chocolate factory where we consider the inputs and outputs at each step in the chain. Students reason about the sequence in which a composition of functions is evaluated and how the derivative of a function can only be taken with respect to its input, and sometimes this input is actually a function itself. It is important to note that we are teaching the chain rule in the “reverse” order, so it may look different than you’re used to seeing in textbooks. We feel that moving from the inside function to the outside function will not only help students avoid mistakes but also build on prior knowledge and support conceptual understanding. See “Student Misconceptions” below for more on this.
The chain rule will take a while for students to fully grasp. The first day’s lesson is focused on students’ conceptual understanding and the second and third day is focused on procedural fluency. As much as possible, pause to have students summarize the key ideas of the lesson: “Why can’t we just take the derivative of the outside and inside functions with respect to x?”
This lesson incorporates some fundamental review of composite functions; depending on the algebra background of your students, you may wish to spend longer or shorter on this portion of the lesson. The CalcMedic approach to the chain rule builds from student understanding of how composite functions are evaluated. When evaluating a composite function of the form h(x) = f(g(x)), students first identified the value of the independent variable, x and used it as the “input” for the inner function, g(x). Then the “output” value, g(x), served as the “input” for the outer function, f.
CalcMedic uses the same order of operations when finding the derivative, h’(x), of a composite function such as h(x) = f(g(x)): have students consider the independent variable, x, first. The derivative of x is dx/dx or 1. Explain that this term describes the change in x compared to the change in x. Because x changes at the same rate as itself, this ratio is 1. The term dx/dx is rarely written in a derivative expression because its value is 1. However, this is the optimum moment to introduce the term as it will reappear during implicit differentiation and when related rates problems are investigated.
The chain rule is all over the AP exam! Found in both the multiple choice and free response questions, students must know how to evaluate the derivative of composite functions and be familiar with all the notation associated with it. Often times, the inside functions and outside functions are not explicitly given and students must demonstrate a deeper conceptual understanding of the chain rule and when to apply it. (Ex: Find the derivative of h(2x))
We have found that students make several common errors when using the chain rule. First, they take the derivatives of the inside function and outside function at the same time, forgetting to “keep the inside function intact”. Second, if they do take the derivative with respect to the inside function, they often forget to then multiply by the derivative of the inside function. This mistake is an answer choice for almost all multiple choice questions involving the chain rule. Thirdly, some students don’t know when to stop taking derivatives. If the derivative of the inside function is 2x, they multiply by an additional factor of 2 because the derivative of 2x is 2.
We believe that teaching the chain rule as g’(x)*f’(g(x)) will greatly mitigate many of these errors. This is because once students take the derivative of the inside function, g(x), they will likely not “reuse” that derivative inside f’. Since they start by taking the derivative of g(x), they are likely not to forget to take its derivative. Additionally, 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!