Overview

Along with the derivative definitions and rules learned so far, the product rule is another foundational algorithm that students will use often throughout the AB and BC course.  The context for this lesson was interesting to our students and we had strong engagement in the lesson. By itself, the product rule is generally not a challenge for calculus students, so we chose to make notational fluency, graphical representations, and connecting representations our focus today. Students are provided a visual explanation of the product rule and are then asked to develop the product rule on their own. 

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Teaching Tips

Review the formalization notes in the margin so you are able to clarify for students the meaning of all regions in questions 2 and 3. Be able to explain why the product of ∆r and ∆w is insignificant in this context. Be aware of how the notation changes throughout the page. The regions in the diagram represents the change in photos, whereas question 4 gets at the rate of change; these are related but not identical. Explain to students that the Leibniz notation uses dr, dw, and dt to refer to a tiny, infinitesimal change. The concept of a limit is embedded in the notation! As a challenge for advanced learners, have students investigate the product of three functions, f(x)∙g(x)∙h(x). The resulting derivative is intuitive and easy to remember!

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Exam Insights

This is a required skill that is tested on its own and as an intermediate step in more complex questions. Found on both the MC and FRQ sections of the test, students will be successful on these questions with consistent exposure to derivatives of products.

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Student Misconceptions

Generally, students can perform the product rule algorithm; simplification of terms and algebraic manipulation is often the greatest challenge. Perfect practice makes perfect.