The following problems require the use of the product rule. In the following discussion and solutions the derivative of a function

.

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 .Click HERE to see a detailed solution to problem 1.

*PROBLEM 2 :*Differentiate .Click HERE to see a detailed solution to problem 2.

*PROBLEM 3 :*Differentiate .Click HERE to see a detailed solution to problem 3.

*PROBLEM 4 :*Differentiate .Click HERE to see a detailed solution to problem 4.

*PROBLEM 5 :*Differentiate .Click HERE to see a detailed solution to problem 5.

*PROBLEM 6 :*Differentiate .Click HERE to see a detailed solution to problem 6.

*PROBLEM 7 :*Differentiate .Click HERE to see a detailed solution to problem 7.

*PROBLEM 8 :*Differentiate .Click HERE to see a detailed solution to problem 8.

*PROBLEM 9 :*Differentiate .Click HERE to see a detailed solution to problem 9.

*PROBLEM 10 :*Differentiate .Click HERE to see a detailed solution to problem 10.

*PROBLEM 11 :*Differentiate .Click HERE to see a detailed solution to problem 11.

*PROBLEM 12 :*Differentiate .Click HERE to see a detailed solution to problem 12.

*PROBLEM 13 :*Consider the function . For what values of*x*is*f*'(*x*) = 0 ?Click HERE to see a detailed solution to problem 13.

*PROBLEM 14 :*Consider the function . For what values of*x*is*f*'(*x*) = 0 ?Click HERE to see a detailed solution to problem 14.

*PROBLEM 15 :*Consider the function . For what values of*x*is*f*'(*x*) = 0 ?Click HERE to see a detailed solution to problem 15.

*PROBLEM 16 :*Prove that.

This is called the

*triple product rule*. Compare it with the ordinary product rule to see the similarities and differences.Click HERE to see a detailed solution to problem 16.

*PROBLEM 17 :*Differentiate .Click HERE to see a detailed solution to problem 17.

*PROBLEM 18 :*Consider the function . For what values of*x*is*f*'(*x*) = 0 ?Click HERE to see a detailed solution to problem 18.

*PROBLEM 19 :*Find an equation of the line tangent to the graph of at .Click HERE to see a detailed solution to problem 19.

*PROBLEM 20 :*Find an equation of the line perpendicular to the graph of at .Click HERE to see a detailed solution to problem 20.

*PROBLEM 21 :*Find all points (*x*,*y*) on the graph of with tangent lines parallel to the line*y*+*x*= 12 .Click HERE to see a detailed solution to problem 21.

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

Fri May 30 13:25:21 PDT 1997