Overview
We have come a long way in just over a week’s time: from the initial basic rules for power and exponential functions, to the structure rules given by the sum and constant multiple rules, then the basic rules for the sine and cosine functions, followed by the more complicated structure rules given by the product and quotient rules. With those rules in hand, we have found derivatives for the remaining trigonometric functions, and we are now ready to deal with one more way that functions can be combined: composite functions, also known as chains of functions. The derivative rule which helps us deal with these is called the Chain Rule.
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.

State, from memory, all of the basic function and structure rules noted in the Overview above.

Be able to accurately state the Chain Rule: \(\frac{d}{dx}[f(g(x))] = \ldots\).

Identify the “inner” and “outer” function in a composite function of the form \(C(x) = f(g(x)),\) such as in \(C(x) = e^{x^2 + 1}\) or \(D(x) = \sqrt{\cos(x) + 4}\).
Advanced learning objectives
In addition to mastering the basic objectives, here are the tasks you should be able to perform after class, with practice:

Apply the Chain Rule to basic examples.

Differentiate complicated functions that require the use of multiple rules at once. For instance, compute the derivative of \(\displaystyle g(x) = \frac{\sin^{2}(x) \cdot e^{x5}} {\sqrt{x^2 + 1}}\).
To prepare for class

Do the Preview Activity for section 2.5 (on WeBWorK if required by your teacher).

Read subsection 2.5.1 in Active Calculus, up to and including the statement of the Chain Rule.

Watch the following videos as needed (the second, third, and fourth videos were created at CCSL):

Watch the following video showing an applied example involving chains of trigonometric functions (created at CCSL):

Do some experimentation with the following interactive applet: Intuitive Notion of the Chain Rule