SUNY Geneseo Department of Mathematics

The Product and Quotient Rules, Part 1

Monday, September 28

Math 221 02
Fall 2020
Prof. Doug Baldwin

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The next session will be 4:30 - 6:00 on Tuesday. Watch for a link announced a hour or so before. This session can focus on problem set 3, and the product and quotient rules.

The Product Rule

To illustrate the basic idea, use the product rule to find a derivative. It’s helpful to identify and name the parts of the product in order to write out the form of the rule. Then plug the functions and their derivatives into that form and simplify:

Using the product rule on a product of polynomials

The textbook uses the fact that the derivative of x2 is not the product of the derivative of x with itself to show that derivatives of products must be more complicated than just the product of the derivatives. Can you do the reverse to check the product rule? In other words, if you use the product rule to calculate the derivative of x · x do you get 2x, the derivative of x2?

Indeed you do, keeping in mind that the derivative of x with respect to itself is always 1:

The product rule finds derivative of x times x to be 2 x

As you no doubt know, the area of a rectangle of width w and height h is hw. Suppose a rectangle is growing in such a way that the change in width with respect to time is 2 inches per second, and the change in height with respect to time is 4 inches per second. How fast is the area changing at the moment the rectangle’s width is 10 inches and height is 15 inches?

The trick here is to use the product rule on function and derivative values at a single point in time, don’t try to guess the actual functions for width and height. Then you use the product rule to tell you which numbers to plug into the product rule formula:

Rectangle with width, height, and rates of change for each; product rule applied to those quantities

(And notice that the units even come out right, i.e., the product rule combines distances and changes in distance per unit of time in such a way as to produce an answer whose units are change in area per unit of time.)

The Quotient Rule

Use the quotient rule to find some derivatives. As with the product rule, it’s helpful to name the parts of the quotient so you can use them in writing out the rule. Then plug in the parts and their derivatives and simplify:

Quotient rule applied to 2 x squared plus 1 all over x minus 2

Here’s another example:

Quotient rule applied to s squared over the quantity s cubed plus 1

Let’s see if we can do a variation on the book’s proof of the power rule for derivatives extended to negative integer powers. In particular, I find that it’s easier to keep track of what’s going on if you try to find the derivative of x-n, where n is a positive integer:

Quotient rule proves derivative of x to the minus n is minus n times x to the minus n minus 1

Next

(Thursday, since Wednesday is a rejuvenation day.)

Look at more examples of the product and quotient rules.

No new reading. But think about the examples and post thoughts, questions, etc. to the product and quotient rule discussion.

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