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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

  1. 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.
  2. Students use their knowledge from the preceding lecture to find the total differentials of given functions.
  3. 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

  1. This small group activity is designed to provide students with a means of experimentally verifying relationships between partial derivative expressions.
  2. 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.
  3. 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

  1. 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.
  2. 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.
  3. 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

  1. This small group activity is designed to show students how to calculate derivatives using small differences while paying attention to what is held constant.
  2. Students use the Partial Derivative Machine to measure partial derivatives while keeping different variables of the system constant.
  3. 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

  1. This small group activity is designed to help students become familiar with the Partial Derivative Machine and how to think about measuring derivatives.
  2. Students practice thinking about which derivatives are easy and hard to measure in the context of the PDM.
  3. 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

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

Homework

  1. (mbTreeDiagramMath) Use Tree Diagrams to find chain rules.

    For each of the sets of functions below, draw a chain rule diagram for the indicated derivative and use it to write a chain rule, then evaluate your chain rule to find the derivative.

    1. Find $\left( \frac{dg}{dt} \right)$ for $g = (a + b)^2$, $a = \sin 2t$, and $b = t^{3/2}$.

    2. Find $\left( \frac{\partial h}{\partial v} \right)_u$ for $h = \sqrt{a - b}$, $a = uv^2 - 1/v$, and $b = \frac{uv}{u + v}$.

    3. Find $\left( \frac{\partial A}{\partial B} \right)_F$ for $A(B,C)$ and $F(B,C)$. (Of course, you don't have to evaluate this derivative!)

  2. (mbCyclic) Check cyclic chain rule for realistic equation of state

    A possible equation of state for a gas takes the form $$pV=N k_B T \exp\left(-\frac{\alpha V}{N k_B T}\right)$$ in which $\alpha$, $N$, and $k_B$ are constants. Calculate expressions for: $$\left(\frac{\partial p}{\partial V}\right)_T\qquad\qquad \left(\frac{\partial V}{\partial T}\right)_p\qquad\qquad \left(\frac{\partial T}{\partial p}\right)_V$$ and show that these derivatives satisfy the cyclic chain rule.

  3. (mbParamagnet) Use chain rules to solve physics problem

    Paramagnetism\hfill\break We have the following equations of state for the total magnetization $M$, and the entropy $S$ of a paramagnetic system: \begin{eqnarray*} M&=&N\mu\, \frac{e^{\frac{\mu B}{k_B T}} - e^{-\frac{\mu B}{k_B T}}} {e^{\frac{\mu B}{k_B T}} + e^{-\frac{\mu B}{k_B T}}}\\ \noalign{\smallskip} S&=&Nk_B\left\{\ln 2 + \ln \left(e^{\frac{\mu B}{k_B T}}+e^{-\frac{\mu B}{k_B T}}\right) +\frac{\mu B}{k_B T} \frac{e^{\frac{\mu B}{k_B T}} - e^{-\frac{\mu B}{k_B T}}} {e^{\frac{\mu B}{k_B T}} + e^{-\frac{\mu B}{k_B T}}} \right\}\\ \end{eqnarray*}

    1. Solve for the magnetic susceptibility, which is defined as: $$\chi_B=\left(\frac{\partial M}{\partial B}\right)_T $$

    2. Also solve for almost the same derivative, now taken with the entropy $S$ held constant: $$\left(\frac{\partial M}{\partial B}\right)_S $$

    3. Why does this second derivative turn out to be zero?

    4. Sense-making: solve explicitly for the chain rule that allows you to evaluate $$\left(\frac{\partial M}{\partial B}\right)_S $$ using both total differentials (zapping with d) and a chain rule diagram. (Your chain rule should be the same!)


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