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## Cyclic Chain Rule: Instructor's Guide

### Main Ideas

- Measuring derivatives on the Partial Derivative Machine (PDM)
- Evaluating dimensions of a cyclical chain rule
- Experimentally verifying a cyclical chain rule using the PDM
- Introducing students to the concept of the cyclical chain rule

### Students' Task

*Estimated Time: 15 minutes*

Students are given the prompt: In your groups verify the correctness of the following equation $\left(\frac{\partial x_2}{\partial F_1}\right)_{x_1} = -\frac{\left(\frac{\partial x_1}{\partial F_1}\right)_{x_2}}{\left(\frac{\partial x_1}{\partial x_2}\right)_{F_1}}$

To verify the correctness of the equation, students are given two tasks:

- Check that the dimensions of the expression make sense
- Experimentally confirm this relationship using the Partial Derivative Machine

### Prerequisite Knowledge

- Familiarity with partial derivatives
- Familiarity with the Partial Derivative Machine

### Props/Equipment

- Tabletop Whiteboard with markers

### Activity: Introduction

### Activity: Student Conversations

- Ensure that students understand that they can “rotate” the differential positions in either direction, so long as they are consistent.

### Activity: Wrap-up

Once all or most of the groups have had a chance to measure each necessary derivative, convene the class and introduce the concept of the cyclical chain rule. If they haven't already, have groups verify the cyclical chain rule with the derivatives they have found. Take some time to discuss the dimensions of the Cyclical Chain Rule, the significance of the relation and how it will be used, as well as how to know when not to use it.

### Extensions

This activity is the second activity of the Partial Derivative Machine (PDM) Sequence on partial derivative relations. This sequence uses the Partial Derivative Machine (PDM).

- Preceding activities:

- Follow-up activities: