### DIFFERENTIATION USING THE PRODUCT RULE

The following problems require the use of the product rule. In the following discussion and solutions the derivative of a function h(x) will be denoted by or h'(x) . The product rule is a formal rule for differentiating problems where one function is multiplied by another. The rule follows from the limit definition of derivative and is given by

.

Remember the rule in the following way. Each time, differentiate a different function in the product and add the two terms together. In the list of problems which follows, most problems are average and a few are somewhat challenging. In most cases, final answers to the following problems are given in the most simplified form.

### The following problems require use of the chain rule.

• PROBLEM 7 : Differentiate .

• PROBLEM 8 : Differentiate .

• PROBLEM 9 : Differentiate .

• PROBLEM 10 : Differentiate .

• PROBLEM 11 : Differentiate .

• PROBLEM 12 : Differentiate .

• PROBLEM 13 : Consider the function . For what values of x is f'(x) = 0 ?

• PROBLEM 14 : Consider the function . For what values of x is f'(x) = 0 ?

• PROBLEM 15 : Consider the function . For what values of x is f'(x) = 0 ?

• PROBLEM 16 : Prove that

.

This is called the triple product rule . Compare it with the ordinary product rule to see the similarities and differences.

• PROBLEM 17 : Differentiate .

• PROBLEM 18 : Consider the function . For what values of x is f'(x) = 0 ?

• PROBLEM 19 : Find an equation of the line tangent to the graph of at .

• PROBLEM 20 : Find an equation of the line perpendicular to the graph of at .

• PROBLEM 21 : Find all points (x, y) on the graph of with tangent lines parallel to the line y + x = 12 .

Duane Kouba
Fri May 30 13:25:21 PDT 1997