The Chain Rule is the following (taken from Sec 3.6 of the textbook)–it shows how to compute the derivative of a composite function *h(x) = f(g(x))*:

The key to using the Chain Rule is to analyze a given composite function in terms of the “outside function” (the function *f* in the notation above) and the “inside function” (*g* above). The Chain Rule says the derivative of the composite function is “the derivative of the outside function evaluated at the inside function” (i.e., *f'(g(x))*) times “the derivative of the inside function” (i.e., *g'(x)*).

Here are a couple examples from the textbook. In Example 3.49, for *h(x) = (sin x)^3*, the “outside function” is the cubing function *f(u) = u^3*, and the “inside function” is *g(x) = sin x*:

What are the “outside” and “inside” functions in the following example?

Here is the Chain Rule as I presented it in class (but note that I wrote it out there applied to h(x) = g(f(x))), along with another “Checkpoint” example from the textbook:

Here are a few more Chain Rule examples we did in class:

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