## Representations of Ordinary Derivatives

Students entering into middle-division physics courses often have a limited working definition of the derivative and tend to consider the derivative an analytic procedure. This sequence introduces the derivative using many different representations and requires that students move between representations of the derivative. Additionally, differences in typical ways which mathematicians and physicists may different in their respective disciplines regarding the derivative can be discussed. This sequence is intended to prompt student thinking using multiple representations including those which may not have been used in introductory physics or calculus courses.

### Activities

• "Derivative Machine" (Estimated time: 20 minutes): This small group activity has students find the relationship between $x$ and $F$ and then students are to determine $\frac{dx}{dF}$. This encourages students to consider that the derivative can be experimentally measured as a ratio of small changes. This activity can be concluded with a homework problem asking students to analyze their data.
• Recall the Derivative (Estimated time: 5-10 minutes): This small whiteboard question prompts students to write something they know about derivatives which yields a variety of responses. These responses initiate a conversation about various representations of derivatives and associated ideas such as examples of time derivatives, graphical representations of derivatives, slope at a point, a rate of change, ratio of small quantities, the limit definition, and “instantaneous”.
• Lecture on Multiple Representations of Derivatives (Estimated time: 10-15 minutes): This lecture can briefly follow up from the SWBQ to extend and remind students of other representations of the derivative which may not have occurred from Recall the Derivative as well as relate the various representations of derivatives. This lecture can relate back to the "Derivative Machine" by discussing how the data could be used to find the derivative. The instructor should encourage students to think about plotting data and determining the derivative numerically. This lecture can be extended by discussing how to deal with experimental data when the theoretical model is unknown.
• Lecture on Derivatives in Physics (Estimated time: 10-15 minutes): This optional lecture describes derivatives as ratios of small changes which pertain to the size scale of interest. The scale in physics is relevant to the particular discipline (i.e. astrophysics works on a much large scale than atomic physics), and therefore the scale must be chosen to represent an experiment in a relevant way.

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