SUNY Geneseo Department of Mathematics

The Chain Rule, Part 2

Thursday, October 8

Math 221 02
Fall 2020
Prof. Doug Baldwin

Anything You Want to Talk About?

Review the book’s use of the Squeeze Theorem to show that lim_{x → 0} x cos x = 0:

The thinking behind that proof starts with noticing that cos x is between -1 and 1. So multiplying x by cosine x is always multiplying x by something whose absolute value is less than or equal to 1 — it always produces a product closer to 0, or at worst as close to 0, as x is. So if you graph the lines y = x and y = -x, x cos x has to lie in between those lines:

Shaded region between graphs of y equals x and y equals minus x

Now if you look at just the positive parts of those 2 lines, you have the graph of y = |x|. Similarly, the negative parts of the lines are the graph of y = -|x|. So the idea that x cos x is between the lines can be expressed a little more formally as -|x| ≤ x cos x ≤ |x|. Then the Squeeze Theorem says x cos x has to have the same limit as -|x| and |x|, which is 0.

x cosine x is between minus absolute value of x and absolute value of x, so limit is 0

Colloquium

(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.

More About the Chain Rule

Based on section 3.6 in the textbook and the chain rule discussion.

Using the Chain Rule with Other Rules

For example, differentiate sin(t²)/cos²t.

This looks like it needs the quotient rule, but another way to approach it is to rewrite it as a product involving cos^-2t:

Rewrite quotient of sine t squared and cosine squared of t as sine of t squared times cosine to the minus 2

Now use the product rule, which is simpler than the quotient rule:

Sum of products each involving a derivative

Finally, find the derivatives needed by the product rule by using the chain rule twice. Each use of the chain rule requires identifying the “f” and “g” functions:

Using the chain rule to find derivatives in a sum of products

Multiple Compositions of Functions

For example, differentiate tan⁵( x³ - 3x² ).

Rewriting the expression slightly to emphasize raising the result of tangent to a power makes it a little easier to see what has to be done:

Tangent to the fifth power of big expression is tangent of big expression all to the fifth

In particular, apply the power rule to the result of the tangent function, but since you’re differentiating the result of a function, you have to use the chain rule:

Using the chain rule to differentiate tangent to the 5th

But now the g′(x) part from the chain rule also requires the chain rule. That’s not a problem…

Using the chain rule inside the result of another use of the chain rule

One more day of practice with the chain rule.

No new reading to do, but do keep looking at the chain rule discussion and seeing if you can contribute to some of the questions that haven’t been discussed yet.

Next Lecture