Section 3.6 Worksheet:

Assigned problems: Exercises pp. 183-185, #2-10 even; 55, 58, 66, 67 (due Wednesday, 2/19)

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 .

5. One explicit (and very important) example of the chain rule is given, for power function. Can you see how to use this to turn the quotient rule into the product rule? How do we compute the derivative of the quotient

   

   using the chain rule for powers together with the product rule?

Notes:

1. The proof of the chain rule (p. 182: How to prove the chain rule....) may be skipped.
2. 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.