This sub-organizer contains the following sections:
Prescribed Learning Outcomes
Suggested Instructional Strategies
Suggested Assessment Strategies
Recommended Learning Resources

PRESCRIBED LEARNING OUTCOMES

It is expected that students will determine derivatives of functions using a variety of techniques.

It is expected that students will:

• compute and recall the derivatives of elementary functions including:
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• use the following derivative formulas to compute derivatives for the corresponding types of functions:constant times a function:
  • constant times a function:
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• use the Chain Rule to compute the derivative of a composite function:
  
  
  
• compute the derivative of an implicit function.
• use the technique of logarithmic differentiation.
• compute higher order derivatives

SUGGESTED INSTRUCTIONAL STRATEGIES

Knowledge of a variety of techniques for computing the derivatives of different types of functions allows students to solve an ever-increasing range of problems. To become efficient problem solvers students need to understand when and how to use derivative formulas, but will also find it useful to memorize some basic derivatives.

• Point out to students that the derivatives of a few functions will already have been obtained using the definition of a derivative.
• To help students connect the limit concept to the derivative of sin x, demonstrate the details of , using the known fact that and that . Alternatively, you can provide these known facts to students and have them work in groups to develop the solution.
• Have students work in groups to differentiate using two different methods:
  1) the chain, product, and quotient rules
  2) logarithmic differentiation
  Ask students to compare the two methods.
• Demonstrate that the derivatives of functions such as can be found using implicit differentiation. For example, given can be determined implicitly as follows:
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• Although full proof of the chain rule is uninformative, a graphing calculator can be used to show quite persuasively that for example ought to be 3 cos 3x at any particular place .

SUGGESTED ASSESSMENT STRATEGIES

As students develop confidence and proficiency computing the derivative of elementary functions, they are better prepared to solve more complex problems that involve derivative formulas. Assessment should focus on students’ ability to recall derivatives of elementary functions and apply this knowledge appropriately when computing the derivative of a more complex function (using formulas, implicit differentiation, or logarithmic differentiation).

Observe

• As students are working on problems, circulate and provide feedback on their notation use. Have students verify their work using a graphing calculator.

Collect

• For a question where ask students to debate the effectiveness of finding the derivative using expansion, the chain rule, the product rule, and logarithmic differentiation. To what extent do their arguments illustrate their ability to expand on current mathematical idea?
• Have students present to the class research on some elementary functions and their derivatives.
• Discuss with students the merits of the various methods of taking derivatives. Have them summarize in writing their understanding of the merits of each. Work with them to develop a set of criteria to assess the summaries.

Self-Assessment

• To assess students’ command of the chain rule, have them list common difficulties they encounter in taking derivatives. Let them work in pairs to develop checklists of reminders to use when they’re checking their work.
• Ask students to summarize derivatives of elementary functions, making notes on notation errors. Allow students to complete assignments using their summaries.

RECOMMENDED LEARNING RESOURCES

Print Materials

• Calculus: Graphical, Numerical, Algebraic
  Ch. 3 (Sections 3.3, 3.5, 3.6, 3.7, 3.8, 3.9)
  pp. 119, 151-154, 169
• Calculus of a Single Variable Early Transcendental Functions, Second Edition
  pp. 121, 125-126, 133-134, 137, 139, 143, 149, 158, 163, 168, 170
• Single Variable Calculus Early Transcendentals, Fourth Edition
  Ch. 3 (Sections 3.1, 3.2, 3.4, 3.5, 3.6, 3.7 3.8)

Multimedia

• Calculus: A New Horizon, Sixth Edition
  pp. 189, 191-196, 200, 204, 246, 257, 258, 261
• Calculus of a Single Variable, Sixth Edition
  Ch. 2 (Section 2.5)
  pp. 103, 105-107, 114-120,125-126, 315-316, 340, 380

© 2000 Copyright. All Rights Reserved. Curriculum Branch.
Maintained by: Mathematics Coordinator
Revised: December 5, 2000

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