Overview
This section covers the following concepts: The product and quotient rules for differentiation.
Basic learning objectives
These are the tasks you should be able to perform with reasonable fluency when you arrive at your next class meeting. Important new vocabulary words are indicated in italics.

(Review) Apply all the derivative rules from Sections 2.1 and 2.2 with fluency.

Explain why the derivative of a product of two functions, \(f(x) \cdot g(x)\), is not just the product of the derivatives, \(f'(x) \cdot g'(x)\). Give a specific example to show that the derivative of a product is not the product of the derivatives.

State the Product Rule and use it in a simple situation.

State the Quotient Rule and use it in a simple situation.
Advanced learning objectives
In addition to mastering the basic objectives, here are the tasks you should be able to perform after class, with practice:

Prove the Quotient Rule using the Product Rule (see the proof in the book in Section 2.3).

Differentiate a function for which the derivative involves a combination of the Product Rule, Quotient Rule, and other rules we’ve learned.

Use the Product and Quotient rules in the context of a realworld problem to find the slope of a tangent line, the instantaneous rate of change in a function, or the instantaneous velocity of an object.
To prepare for class

Do the Preview Activity for section 2.3 (on WeBWorK if required by your teacher).

Read subsection 2.3.1 in Active Calculus about the Product Rule (without doing the activity). Then read subsection 2.3.2 in Active Calculus about the Quotient Rule.

Watch the following videos which show examples where the Product Rule and the Quotient Rule are used (try to do the examples shown in the videos yourself, by pausing the video before it shows the solutions!):

Watch the following video which shows how we can combine the different rules that we have seen so far: the constant multiple rule, the sum/difference rule, the derivatives of polynomial, exponential, sine, and cosine functions, the product and the quotient rule:

Watch the following video which gives an explanation of the Product Rule using an example from economics (video created at CCSL):

Watch the following video which shows how to use the Product Rule to compute the derivative at a specific position when only graphical representations of the functions are given (video created at CCSL):