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## Upside Down Derivatives: Instructor's Guide

### Main Ideas

- This small group activity is designed to provide students with a means of experimentally verifying relationships between partial derivative expressions
- Students use the Partial Derivative Machine (PDM) to measure two “easy” derivatives that are mathematical reciprocals of each other in order to demonstrate a relationship between them.
- The wrap up discussion focuses on helping students realize that when the variables in the numerator and denominator of a partial derivative are switched, and the same variable is held constant, that the numerical value of the derivative is simply the reciprocal of the original quantity; which was hopefully demonstrated by the student measurements.

### Students' Task

*Estimated Time: 10 minutes*

Using your devices, measure the following two derivatives: $\left(\frac{\partial x_1}{\partial x_2}\right)_{F_1}$ and $\left(\frac{\partial x_2}{\partial x_1}\right)_{F_1}$

How do they relate to one another?

### Prerequisite Knowledge

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

### Props/Equipment

- Tabletop Whiteboard with markers
- A Partial Derivative Machine per group

### Activity: Introduction

### Activity: Student Conversations

### Activity: Wrap-up

The wrap up discussion for this activity is to have students report the numerical values they found and encourage them to realize that the derivatives should be related by the expression: $\left(\frac{\partial x_1}{\partial x_2}\right)_{F_1} = \frac{1}{\left(\frac{\partial x_2}{\partial x_1}\right)_{F_1}}$

### Extensions

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

- Follow-up activities: