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# Derivatives and Chain Rules

## Prerequisites

Students should be able to:

- Recognize the traditional calculus notation for partial derivatives (
*i.e.,*$\partial f/\partial x$). - Calculate a partial derivative given a symbolic expression and the variable(s) to be held constant.
- Determine a total differential by zapping with d.
- Write the thermodynamic identity for both the Partial Derivative Machine and for a gas in a piston.
- Reason about the physical quantities related to the Partial Derivative Machine.

## In-class Content

### Activity: Calculating a Total Differential

Link to Calculating a Total Differential Activity

**Activity Highlights**

- This small group activity is designed to give students a chance to exercise their newly learned skills in taking total differentials of multi-variable functions.
- Students use their knowledge from the preceding lecture to find the total differentials of given functions.
- The whole class discussion focuses on good habits to have while calculating total differentials of complicated functions.

### Activity: Upside Down Derivatives

Link to Upside Down Derivatives Activity

**Activity Highlights**

- 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.

### Activity: Cyclic Chain Rule

Link to Cyclic Chain Rule Activity

**Activity Highlights**

- This small group activity is designed to help students practice measuring derivatives with the Partial Derivative Machine (PDM) and help them make connections between mathematical expressions and physical systems and measurement.
- Students are given a written representation of the Cyclic Chain Rule and are asked to assess whether the dimensions make sense and to use the PDM to verify the expression.
- The whole class discussion focuses on the results of the measurements and leads to the instructor introducing the concept of the “Cyclic Chain Rule”.

### Activity: Isowidth and Isoforce Stretchability

Link to Isowidth and Isoforce Stretchability Activity

**Activity Highlights**

- This small group activity is designed to show students how to calculate derivatives using small differences while paying attention to what is held constant.
- Students use the Partial Derivative Machine to measure partial derivatives while keeping different variables of the system constant.
- The whole class discussion focuses on how to represent derivatives in multiple ways, experimentally measure derivatives, and show that the “thing held constant” both has physical and mathematical consequences.

### Activity: Easy and Hard Derivatives

Link to Easy and Hard Derivatives Activity

**Activity Highlights**

- This small group activity is designed to help students become familiar with the Partial Derivative Machine and how to think about measuring derivatives.
- Students practice thinking about which derivatives are easy and hard to measure in the context of the PDM.
- The whole class discussion focuses on becoming familiar with the PDM, how to think about derivatives, and which derivatives are easy to measure and which are hard.

### Activity: Dividing by Differentials

Link to Dividing by Differentials Activity

**Activity Highlights**

- This small group activity is designed to ensure that students know when they can (or can't) divide by differentials.
- Students decide in which of the given situations dividing by a differential is a legal move.
- The whole class discussion focuses on how difficult it is to consistently use differential division correctly.

### Lecture: Clairaut's Theorem

### Lecture: Chain Rule Diagrams

Chain Rule Diagrams (Lecture: 20 min)

### Activity: New Surfaces (Squishability of Water Vapor

New Surfaces activity - "Squishability" of Water Vapor (SGA - 20 min)

### Homework

- (mbTreeDiagramMath)
*Use Tree Diagrams to find chain rules.* - (mbCyclic)
*Check cyclic chain rule for realistic equation of state* - (mbParamagnet)
*Use chain rules to solve physics problem*