The following problems require the use of the chain rule. The chain rule is a rule for differentiating compositions of functions. In the following discussion and solutions the derivative of a function h(x) will be denoted by tex2html_wrap_inline53 or h'(x) . Most problems are average. A few are somewhat challenging. The chain rule states formally that

tex2html_wrap_inline57 .

However, we rarely use this formal approach when applying the chain rule to specific problems. Instead, we invoke an intuitive approach. For example, it is sometimes easier to think of the functions f and g as ``layers'' of a problem. Function f is the ``outer layer'' and function g is the ``inner layer.'' Thus, the chain rule tells us to first differentiate the outer layer, leaving the inner layer unchanged (the term f'( g(x) ) ) , then differentiate the inner layer (the term g'(x) ) . This process will become clearer as you do the problems. In most cases, final answers are given in the most simplified form.

The following seven problems require more than one application of the chain rule.

The following three problems require a more formal use of the chain rule.

Click HERE to return to the original list of various types of calculus problems.

Your comments and suggestions are welcome. Please e-mail any correspondence to Duane Kouba by clicking on the following address :

Duane Kouba
Tue May 6 17:21:40 PDT 1997