SOLUTION 9

DIFFERENTIATION USING THE CHAIN RULE

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 or h'(x) . Most problems are average. A few are somewhat challenging. The chain rule states formally that .

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.

• PROBLEM 1 : Differentiate .

• PROBLEM 2 : Differentiate .

• PROBLEM 3 : Differentiate .

• PROBLEM 4 : Differentiate .

• PROBLEM 5 : Differentiate .

• PROBLEM 6 : Differentiate .

• PROBLEM 7 : Differentiate .

• PROBLEM 8 : Differentiate .

• PROBLEM 9 : Differentiate .

• PROBLEM 10 : Differentiate .

• PROBLEM 11 : Differentiate .

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

• PROBLEM 12 : Differentiate .

• PROBLEM 13 : Differentiate .

• PROBLEM 14 : Differentiate .

• PROBLEM 15 : Differentiate .

• PROBLEM 16 : Differentiate .

• PROBLEM 17 : Differentiate .

• PROBLEM 18 : Differentiate .

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

• PROBLEM 19 : Assume that h(x) = f( g(x) ) , where both f and g are differentiable functions. If g(-1)=2, g'(-1)=3, and f'(2)=-4 , what is the value of h'(-1) ?

• PROBLEM 20 : Assume that , where f is a differentiable function. If and , determine an equation of the line tangent to the graph of h at x=0 .

• PROBLEM 21 : Determine a differentiable function y = f(x) which has the properties and .