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# Partial Derivative Machine

**Keywords:**Derivative, Partial Derivative, Change of Variables, PDM, Small Group Activity

### Highlights of the activity

- This small group activity is designed to provide students with a means of measuring derivatives using ratios of small changes.
- Students measure partial derivatives on the “Partial Derivative Machine.”
- The whole class discussion focuses on how to represent partial derivatives in multiple ways and how to experimentally measure derivatives.

### Reasons to spend class time on the activity

This activity occurs early in the junior year during “Symmetries and Idealizations” when students have already been introduced to derivatives in introductory physics and calculus classes. Students are likely to have a limited working definition of the meaning of a derivative. Most students will primarily think of derivatives as operations on symbolic expressions but will also be familiar with some other representations of the derivative such as the slope of the tangent line at a point in a function which are taught in calculus courses. The derivative as a measurement in an experiment may be familiar from introductory physics courses in the relationships between distance, velocity, and acceleration, however, it is likely few students will immediately grasp this particular definition of a derivative especially when done without time. This activity introduces the derivative as a measurable quantity through taking the ratio of small changes which students can control. Using this particular system, the “Partial Derivative Machine”, students can gain a mechanical intuition of how to measure partial derivatives while avoiding a more complicated physical situation.