Upside Down Derivatives
Highlights of the activity
- 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.
Reasons to spend class time on the activity
This activity occurs during the “Interlude”, a brief course on the basic mathematics used in the “Energy and Entropy” Paradigm. This course primarily focuses on an introduction to partial derivatives and total differentials. We would like our students to have an understanding of how to measure partial derivatives, as well as understand that the “thing held constant” has both physical and mathematical consequences. This activity uses the Partial Derivative Machine (PDM) to demonstrate to students that the result of a derivative being flipped upside-down is known, and using the physical system helps demonstrate that the math is in fact true for physical systems.