### 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.

• PROBLEM 1 : Differentiate .

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• PROBLEM 2 : Differentiate .

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• PROBLEM 3 : Differentiate .

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• PROBLEM 4 : Differentiate .

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• PROBLEM 5 : Differentiate .

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• PROBLEM 6 : Differentiate .

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

• PROBLEM 7 : Differentiate .

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• PROBLEM 8 : Differentiate .

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• PROBLEM 9 : Differentiate .

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• PROBLEM 10 : Differentiate .

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• PROBLEM 11 : Differentiate .

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• PROBLEM 12 : Differentiate .

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

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

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

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• PROBLEM 16 : Prove that .

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

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• PROBLEM 17 : Differentiate .

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

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

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

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

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### 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
Fri May 30 13:25:21 PDT 1997