## Lecture ( minutes)

Give the students a = 6bc - eF^{b/7} and ask them to write down a partial derivative they can find from this equation. Students are likely to come with derivatives of a, and should be encouraged to note what they are holding constant explicitly. They may not realize that it is possible to find inverses of those derivatives, or that it is even possible to find a derivative of b with respect to c. If no students do, ask them to zap with d. This is an opportunity to give an initial glimpse of the cyclic chain rule using differential substitution.

Give the students q = a^{3} + b^{4} + c^{5} and again ask them to write down a partial derivative they can find from this equation. Note that it is critical to note that two things are being held constant in the obvious easy derivatives. Again try zapping with d and discussing the total differential as a new representation of the same information.

Ask students what they would do if they wanted to find the derivative of q with respect to a holding only c constant. Let them think a little and then go through a parallel problem-solving process using three different representations:

(1) symbolic functions, which uses substitution. How do you know which variable to substitute? If you can't substitute, you will need a different method.

(2) differentials. Write out the relationship between total differentials in general, abstract form (with blank spaces before the differentials, or with known partial derivatives). Again, what is it that you want to substitute? Identify the total differential expression you want in the end to get to the desired derivative and perform the necessary substitution to find the chain rule directly.

(3) tree diagrams. Introduce this new representation, using differentials connected by lines to show variable relationships, where the lines are branches that correspond to the partial derivatives. Explicitly describe how to translate between a differentials representation and a tree diagram. Then, show how to combine two tree diagrams for relationships that you have into a single diagram for the relationship that is needed, and how to read the diagram to find the chain rule, showing that it is the same chain rule as before.

An important note about rotating tree diagrams (not covered in 2017): Rotating a chain rule diagram is always a valid thing to do, but it can be hard to read. The proper way to read a diagram that is oriented incorrectly is to use the cyclic chain rule. When in doubt, always write out the total differentials and solve for the variable you want to substitute.