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## Easy and Hard Derivatives: Instructor's Guide

### Main Ideas

The goals of this activity are to help students:

- Understand the apparatus
- Practice thinking about derivatives
- Practice thinking about how to measure a derivative and what constitutes and “easy” or “hard” thing to measure

### Students' Task

*Estimated Time: 10 minutes *

Write down all the derivatives that you could conceivably measure. Which of these are “easy” to measure, and which are “hard?” Why?

### 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

In order to gain practice thinking about measuring derivatives, students are asked to brainstorm all the (partial) derivatives they can think of related to the four physical variables of the Partial Derivative Machine. Once this task has been completed, they categorize the derivatives based on whether or not they are easy to measure. For example the partial derivative $\left(\frac{\partial x_1}{\partial F_1}\right)_{x_2}$ is an easy derivative to measure, you measure the change in $x_1$ as $F_1$ varies at a constant $x_2$. The derivative $\left(\frac{\partial F_2}{\partial F_1}\right)_{x_2}$ is not easy to measure, because the force in a given string cannot be measured if that string is pinned.

### Activity: Student Conversations

### Activity: Wrap-up

Once the class has had a few minutes to come up with partial derivatives, convene the class and have each group report a derivative. Record these derivatives on the white board. Once every group has reported a derivative, cycle through the groups again until every unique derivative has been reported. If there are any partial derivatives that no group found, write them on the whiteboard. Go through the derivatives one by one, having the class say whether they think each derivative is easy or hard, and then describing why each is such before moving on to the next.

### Extensions

This activity is the third activity of the Partial Derivative Machine (PDM) Sequence on measuring partial derivatives and potential energy. This sequence uses the Partial Derivative Machine (PDM).

- Preceding activities:
- Quantifying Change: This small group activity introduces students to the PDM by asking them to determine how many measurable quantities exist within the system and how many of these quantities are simultaneously controllable.
- Isowidth and Isoforce Stretchability: In this small group activity, students are challenged to measure a given partial derivative with the PDM.

- Follow-up activities:
- Legendre Transforms on the PDM: In this small group activity, students get a chance to work with physical analogues of Legendre transforms.
- Potential Energy of an Elastic System: In this integrated laboratory activity, students use the PDM to determine the change in potential energy between two states of a nonlinear system.