SUNY Geneseo Department of Mathematics
Friday, October 9
Math 221 02
Fall 2020
Prof. Doug Baldwin
(No.)
(i.e., a talk by another mathematician sharing something they work on)
Next Tuesday (October 13), from 4:00 to 5:00 via Zoom.
“The Column-Row Factorization A = CR”
Prof. Gilbert Strang, MIT
About teaching linear algebra, but it’s advertised as accessible to everyone.
Prof. Strang is one of the main authors of your textbook.
Sunday 6:00 - 7:30.
The plan is to review this past week, plus talk about the current problem set or other questions, Mathematica, etc.
From section 3.6 in the textbook and the chain rule discussion.
Chain rule problems often involve multiple uses of it. For example, use the chain rule to find the derivative of f(x) = 1 / ( sin(x2) + cos(x2) ).
Since I find the chain rule easier to use than the quotient rule, I’d start by rewriting the function to use a negative exponent instead of a fraction. Then we’ll need the chain rule to deal with the exponent, but we won’t need the quotient rule at all:
Now we need to start using the chain rule, first to differentiate a complicated function (sin(x2) + cos(x2)) raised to the -1 power:
The next derivative we need to take will be a sum of derivatives of sine and cosine of x2. We need the chain rule again to evaluate those derivatives:
Finally, we can do a bit of simplifying to make the result easier to read:
Or consider how the chain rule appears in the distance-to-Mars problem. (I don’t intend to do the entire derivative of the very messy expression; the main value of this example is that it gives you some sense of how the chain rule helps with “real” problems. The example really is a tolerably accurate model of the distance from Earth to Mars over a several-day period, and it really comes from a problem where some students needed its derivative. The chain rule discussion says a little more about the background of this problem.)
The first step in taking the derivative is to apply the chain rule with the square root. It might help to remember that the square root of u is the same as u1/2, so the formula for the derivative of a square root comes from the power rule — in a week or two we’ll see why the power rule does in fact work for fractional exponents:
Evaluating the derivative in the numerator requires more applications of the chain rule, which themselves require even more, for a total of about 7.
Leibniz notation (the notation in which derivatives look like fractions, e.g., dy/dx) makes the chain rule particularly easy to remember, and helps to suggest what it is doing. Think of g(x) in f(g(x)) producing an intermediate variable, say u, and f producing a final result variable, say y:
Then writing the chain rule for the derivative of y looks remarkably like multiplying fractions with a numerator and denominator that cancel out:
So you can remember the chain rule as splitting a fraction into a product of 2 fractions with cancellation, and in some sense the chain rule “works” because of that cancellation (one of the nice things about Leibniz notation is that the metaphor of derivatives as fractions, namely an infinitesimally small change in y over an infinitesimal change in x, while it’s not exactly how modern mathematics thinks about derivatives, is a metaphor that comes very close to reality most of the time).
Extend the set of functions we can differentiate (and find antiderivatives of) by finding derivative rules for exponential functions.
Read “Derivative of the Exponential Function” in section 3.9 of our textbook by class time tomorrow, and participate in this discussion of exponential functions’ derivatives.