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## Finding a Chain Rule: Instructor's Guide

### Main Ideas

This activity's prompt has varied over the years. See below.

#### 2012 (Manogue):

##### Prompt:

Given: $f(x,y)$, $x(t)$, and $y(t)$

Use chain rule diagrams to write the differential of $f$, $df$, in terms of $t$.

Note: this prompt is also used in the ”Calculating a Total Differential” activity.

##### Features:
• abstract math land, although students may interpret this as some function in terms of time-dependent positions
• function of two variables which are both functions of the same variable
• finding total differentials
• using chain rule diagrams
##### Video(s):

\\PARADOCS-NAS\videos\PH423\2012ph423\main\ee12040302mpt2.mov @ 11:20-14:10 (2 minutes, 50 seconds)

#### 2011 (Manogue):

##### Prompt:

Given: $f(x,y)$, $x=x(u,v)$, and $y=y(u,v)$

Using chain rule diagrams, find a chain rule for: $\left (\frac{\partial f}{\partial u}\right )_v$ and $\left (\frac{\partial f}{\partial v}\right )_u$

##### Features:
• abstract math land
• function of two variables which are both functions of (the same) two other variables
• finding chain rules
• using chain rule diagrams

#### 2016 (Manogue):

##### Prompt:

Given: $A(B,C)$ and $C(B,D)$

Find a chain rule for:$\left (\frac{\partial A}{\partial B}\right )_D$ and $\left (\frac{\partial A}{\partial D}\right )_B$

##### Features:
• abstract math land
• two functions of two variables, both functions sharing one variable in common
• finding chain rules
##### Video(s):

\\PARADOCS-NAS\videosnew\PH423\Spring2016\Interlude\main\in16042104mpt1.mov @ 27:05-END (2 minutes, 41 seconds)

\\PARADOCS-NAS\videosnew\PH423\Spring2016\Interlude\main\in16042104mpt2.mov @ 0:00-11:45 (11 minutes, 45 seconds)

#### 2013 (Roundy):

##### Prompt:

Given all other derivatives; $\left (\frac{\partial F_x}{\partial x}\right )_y$, $\left (\frac{\partial F_x}{\partial y}\right )_x$, etc.

How would you find: $\left (\frac{\partial F_x}{\partial x}\right )_{F_y}$ and $\left (\frac{\partial F_x}{\partial F_y}\right )_x$

Note: this prompt is also used in the ” Deriving Change of Variables” activity.

##### Features:
• PDM land
• the PDM is the only given “function;” students are expected to create their own functions from their knowledge of the PDM
• students asked how to find the desired partial derivatives
##### Video(s):

\\PARADOCS-NAS\videos\PH423\2013ph423\main\ee13040404mpt3.mov @ 3:00-36:30 (33 minutes, 30 seconds)

#### 2017 (Emigh):

##### Prompt:

Given: $R=-Tln(S)$ and $U=sin(TS)$

Find a chain rule for: $\left (\frac{\partial U}{\partial T}\right )_R$

##### Features:
• quasi-thermo land
• two functions of the same two variables
• finding chain rules
##### Video(s):

\\PARADOCS-NAS\videosnew\PH423\Autumn2017\ee17110205mpt4.MPG @ 25:05-END (5 minutes, 12 seconds)

### Prerequisite Knowledge

• familiarity with total differentials
• familiarity with methods of finding chain rules
• familiarity with chain rule diagrams
• familiarity with differential substitution

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