Section 3.6 Worksheet:

Assigned problems: Exercises pp. 183-185, #2-10 even; 55, 58, 66, 67 (due Friday)

1. Write the chain rule in two different ways.

2. Which is the ``inner function'' and which the ``outer function'' in the chain rule?

3. What happens if we have a composition of three functions? How does the chain rule change?

4. Identify the inner and outer functions in the composition .

Notes:

1. In section 3.3 we learned differentiation rules for the elementary functions, the building blocks. More complicated functions are often built of compositions of these elementary functions, and the chain rule is the secret to differentiating compositions of functions. This is a terribly important rule which you must memorize and understand.

   The hardest thing about the chain rule is probably identifying the composition of functions. Given an expression, e.g. , you need to realize that , and g(x)=2x-1 (then apply the rule correctly, of course:

   

   Sometimes we talk about ``outer function'' and ``inner function''. The inner function is the first function x meets on its composition voyage. The inner function returns a value u, which serves as the input to the outer function which returns a value y. The composite function of inner and outer thus takes a value x and returns a value y.