The Chain Rule

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Misc

Mid-Semester Feedback Survey

So far it looks like only 1 person has looked at the survey, and they chose not to answer any of the questions. I’m grateful that one of you looked at the survey, but don’t have anything to respond to yet. But I’ll keep checking.

General Education Focus Groups

One of the things I do besides teach calculus is work on Geneseo’s general education program (i.e., the thing that used to require you to take an R/ course and 2 N/ courses and 2 S/ and so forth, and that now asks you to take a quantitative reasoning course and a scientific reasoning course and 3 - 5 “participation in a global society” courses and so forth).

Right now I’m interested in language and approaches to help students see why that requirement is meaningful to them, and to help with that I’m conducting a series of focus group discussions next week — i.e., semi-open-ended discussions with small groups of students designed to explore their reactions to commonly used justifications for general education programs, and to hear their thoughts about Geneseo’s program.

These discussions also feature pizza and soda/water, courtesy of the provost’s office.

Apart from food, you might come out of these discussions with a better understanding of general education at Geneseo and of so-called “liberal education.” Each discussion will last about an hour.

If helping out with this sounds interesting, go to the survey to learn more and/or to indicate times you could participate.

The Chain Rule for Partial Derivatives

Based on section 3.5 in the textbook.

Key Ideas

Derivatives of functions applied to other functions are sums of products of derivatives with respect to intermediate variables times derivatives of intermediate variables.

Examples

Suppose f(x,y) = x sec y, x(t) = 3t², and y(t) = ln t. Use the chain rule to find df/dt.

We started with the chain rule formula for this case:

Then we calculated the various derivatives it calls for and plugged them in to the formula:

Finally, we expressed x and y in terms of t, so there was only 1 variable in the derivative:

Draw a tree diagram for what you just did.

A tree diagram associates derivatives with parts of the structure of a function. It can help either figure out what you need to do to use the chain rule, or document afterwards what you did.

Suppose g(x,y) = xy, x(u,v,w) = u + 2v + 3w, y(u,v,w) = u²v²w². What derivatives does g have with respect to u, v, and/or w, and what are they?

This time g has 3 derivatives, one with respect to each of u, v, and w:

Problem Set

…on various aspects of partial derivatives, including the limit definition, higher-order derivatives, tangents, the chain rule, etc.

Work on it this week and grade it next week.

See the handout for details.

Next

Partial derivatives tell you how fast a function changes as you move a small distance parallel to one of the axes. What if you want to move at an angle to the axes instead?

Then you need so-called “directional derivatives.”

Please read “Directional Derivatives” in section 3.6 of the textbook.

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