Activity: Chain Rules

Prerequisite concepts

Differential Calculus The derivative is a ratio of small changes

One can view $\frac{df}{dx}$ as approximately given by a fraction where $df$ is a small change in $f$ and $dx$ is a small change in $x$.

The derivative is a function

One of Zandieh's process-object layers for derivatives is that derivatives are functions. This means that the value of a derivative depends on where in the domain of the function one is looking. The derivative of a function is itself a function, with the same domain as the original function. Both the (derivative) function and the value of the derivative at a point are commonly referred to as "the derivative."

Variables can be held constant

When taking a derivative of a multivariable function, it is possible to consider a simpler situation in which some of the variables are considered to be constant.

Multivariable Calculus Derivatives can be found while holding one or more variables constant

Derivatives of multivariable functions can be found by holding one or more variables constant (subject to physical limitations).

A partial derivative can be expressed in terms of other partial derivatives

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Zapping with d

Differentials are small changes or differences

One typical notation used for derivatives is Leibniz notation ($e.g.,$ $df/dx$). Physicists often think of the $df$ and $dx$ as distinct quantities (``differentials'') that can be measured, calculated, and manipulated independent of one another. In particular, a differential can be thought of as the (small) change in a quantity when a small change is made to a physical system. A differential can also be thought of as the difference between the values of a quantity between two different (nearby) physical states. This line of thinking is also valuable for thinking about integration in physical situations.

Equations can be 'zapped with d' to relate differentials

FIXME: Does the difference between zapping with d and taking the total differential need to be articulated?

Individual differentials can be manipulated algebraically

FIXME: What? One can do algebra with differentials and reinterpret them as partial derivatives. Can treat like a variable

Total differentials are linear

Partial Derivative Machine Derivatives

Partial derivatives that do not have the same variable(s) held constant (at the same values?) are not the same derivative

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The coefficients in a differentials equation are partial derivatives

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There are experimental limits to how $small$ of a change can be measured The derivative can be interpreted physically

Instructor guide incycchainrule

Chain Rules

FIXME: NOT CORRECT DESCRIPTION: In this activity, students experimentally determine various derivatives using the partial derivate machine, a mechanical analogue for thermodynamic systems. Students explore the ratio, limit, and function aspects of multi-variable derivatives, with an emphasis on holding different variables constant. This activity is also an excellent exercise in representational fluency, as students must coordinate experiments and tables of data with (new) symbolic notations.

Representations used

$df$

Concepts taught

There might be experimental limits on which quantities you can measure

While it may be possible to conceptualize a particular derivative, such as $\left(\frac{\partial U}{\partial S}\right)_{V},$ Where $U$ is internal energy, $S$ is entropy, and $V$ is volume, that does not mean that the particular derivative can be directly measured. In the above example, the entropy $S$ is not measurable.

Differentials allow the finding of partial derivatives when a variable cannot be solved for

Because total differentials are linear, one can find the total differential and then solve algebraically to obtain partial derivatives that might not be obtainable though differentiation.

You can flip a partial derivative if the same variable(s) are constant

$\left(\frac{\partial f}{\partial x}\right)_y$

In other words,[\left(\frac{\partial y}{\partial z}\right)_z = \frac1{\left(\frac{\partial x}{\partial y}\right)_z}] Note that this equality is only true if the same variables are being held fixed on each side of the equality. This therefore relies on the thermodynamics notation that specifies what quantities are being held fixed for a partial derivative.