Module 10  Rules of Differentiation 
Introduction  Lesson 1  Lesson 2  Lesson 3  Lesson 4  SelfTest 
Lesson 10.2: The Product Rule 
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.
Exploring the Derivative of x^{n} sin x 10.2.1 Use your TI89 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 TI89. Click here for the answer. Finding Equivalent Expressions The TI89 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.
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 x^{n} in 10.2.4 Find the derivatives and by using the TI89. 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 TI89. 10.2.5 Predict the generalized derivative and test your prediction with your TI89. 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 TI89. 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.
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. 
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