Lesson 9: 2-9 The Quotient Rule
Duration of Days: 2
Lesson Objective
Students should be able to understand the Quotient Rule as a formula for finding the derivative of a function that is the quotient of two differentiable functions.
Students should be able to state the Quotient Rule from memory, apply the Quotient Rule to find the derivative of a quotient of functions.
Simplify the derivative of a quotient using algebraic techniques.
Recognize when to use the Quotient Rule versus other differentiation rules.
Apply the Quotient Rule in conjunction with other differentiation rules to find derivatives of more complex functions.
What is the Quotient Rule, and when do we use it?
How does the Quotient Rule differ from the Product Rule?
What are the common mistakes students make when applying the Quotient Rule?
How can we remember the formula for the Quotient Rule?
Can you provide an example of a function that requires the use of the Quotient Rule?
Quotient Rule
Numerator
Denominator
Rational Function
Square of the Denominator
Difference of Products
Undefined Derivative
C-D.1: Define the derivative of a function at a point as the limit of a difference quotient. Interpret the derivative as an instantaneous rate of change.
C-D.2: Use derivatives to solve problems involving rates of change.
C-D.3: Find the derivatives of functions using the sum, product, and quotient rules.
This lesson introduces the formula for differentiating a fraction of two functions. Students practice the "Low D-High minus High D-Low" algorithm. Emphasis is placed on maintaining the squared denominator and correctly distributing negative signs during the simplification of the numerator. Like the previous unit, students will also apply this rule to data sets provided in tables.
Purpose: To enable students to differentiate rational functions and functions where variables exist in both the numerator and denominator.
DOK Level 1 (Recall): Reciting the Quotient Rule formula accurately.
DOK Level 2 (Skill/Concept): Differentiating a rational function and simplifying the numerator.
DOK Level 3 (Strategic Thinking): Determining the x-values where a rational function has a horizontal tangent (where the numerator of the derivative equals zero).
DOK Level 4 (Extended Thinking): Deriving the Quotient Rule using the Product Rule and the Chain Rule.
For Struggling Learners (Scaffolding):
Mnemonic Device: Teach the "Ho-De-Hi minus Hi-De-Ho" or "Low D-High" rhyme to prevent students from swapping the terms in the numerator.
For Advanced Learners (Extension):Efficiency Challenge: Give students functions and ask them why the Quotient Rule is "overkill" and how the Power Rule is faster.
Tangent Intercepts: Find the x-intercept of the tangent line to a rational function at a given point.
AP College Board Classroom Assessments