In this lesson you will use an inductive method to develop the rule for the derivatives of products of functions. You will also learn how to save the commands in the History Area as a script. Because you want to save all the commands you enter during the course of this lesson, do not clear the History Area until the end of this lesson.

• Execute the NewProb command from the F6:Clean Up menu.
  Then clear the NewProb command from the History.

Exploring the Derivative of xⁿ sin x

10.2.1 Use your TI-89 to evaluate the derivatives , , and . Click here for the answer.

Predicting the Result and Testing

10.2.2 Predict then use your calculator to test your prediction. Click here for the answer.

Generalizing the Results

10.2.3 Generalize the result by predicting and then test it with your TI-89. Click here for the answer.

Finding Equivalent Expressions

The TI-89 result probably looks different than your prediction. You can determine if the two expressions are equivalent by using the Expand feature of the Algebra menu.

• Clear the Edit Line by pressing
• Paste expand( into the Edit Line by selecting 3:expand( from the :Algebra menu
• Highlight the last result shown in the History Area by pressing
• Paste the last result into the Edit Line by pressing
• Enter the closing parenthesis
• Evaluate the command by pressing

Was this your prediction?

Exploring the Derivative of Other Products

To further explore the derivative of the product of two functions, see what happens if you replace xⁿ in y = xⁿ sin x with other functions and find their derivatives.

10.2.4 Find the derivatives and by using the TI-89. Click here for the answers.

Generalizing the Results

In order to determine the pattern, it's helpful to recall that and that . Both derivatives may be obtained by using the TI-89.

10.2.5 Predict the generalized derivative and test your prediction with your TI-89. Click here to check your answer.

Extending to Other Functions

You have explored the derivative of the product of a general function and sin x. What happens if you replace sin x with other functions?

10.2.6 Predict the derivatives and . Click here for the answers.

Writing the Theorem

10.2.7 Write a theorem that predicts the generalized derivative of the product of two functions, . Then test your theorem with your TI-89. Click here for the answer.

Saving Derivatives

You have examined many derivatives to develop the Product Rule. You can save the commands used to find these derivatives in a script for future use.

• Open the Tools menu by pressing
• Highlight 2:Save Copy As by pressing

• Open the SAVE COPY AS dialog box by pressing
• Move the cursor to the Variable box by pressing
  You will enter the name of the text file in this box.
• Enter the name "prodrule" in the Variable box
  You do not have to use the key since the TI-89 is already in the alpha mode in this box.

• Save the name by pressing
• Close the dialog box and save the file by pressing

The file you just created is called a script and it contains all the commands you entered in this lesson. You will use this script in Lesson 10.3.