Lesson 8: 2-8 The Product Rule
Duration of Days: 3
Lesson Objective
Students should be able to state and correctly apply the product rule to find the derivative of a product of two differentiable functions.
Students should be able to differentiate a variety of functions using the product rule, including polynomial, trigonometric, exponential, and logarithmic functions.
Students should be able to identify when the product rule is necessary to find the derivative of a given function.
How do we identify the two functions, u and v, in a product of functions?
What is the correct order of applying the Product Rule formula?
How can we simplify the derivative after applying the Product Rule?
Can we use the Product Rule in conjunction with other derivative rules, such as the Power Rule or Chain Rule?
Product Rule
Factor
Differentiable
Analytical Representation
Numerical Representation (Table)
Slope of the Tangent
Leibniz Notation
F-IF.4: For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.
This lesson introduces the formal rule for differentiating the product of two functions. Students often struggle with the "common sense" error of simply multiplying f'(x) and g'(x). To correct this, the lesson focuses on the structure: "Derivative of the first times the second, plus the first times the derivative of the second." Practice includes multiplying polynomials by trigonometric functions and working with data tables, which is a high-frequency AP exam task.
Purpose: To expand the differentiation toolkit to functions that cannot be easily expanded or simplified into a single polynomial string.
DOK Level 1 (Recall): Reciting the Product Rule formula and applying it to simple products.
DOK Level 2 (Skill/Concept): Identifying f(x) and g(x) in a complex expression and finding their derivatives separately before assembling the final answer.
DOK Level 3 (Strategic Thinking): Using the Product Rule to find the slope of a tangent line at a specific point for a function like h(x) = f(x)g(x) using a table of values.
For Struggling Learners (Scaffolding):
Color-Coded Matching: Use blue for the "left" function and red for the "right" function to show how they "swap" derivatives in the formula.
For Advanced Learners:
Geometric Proof: Use an area model of a rectangle with sides u and v to show that the change in area.
AP College Board Classroom Assessments