Lesson 2: 2-2 Defining the Derivative
Duration of Days: 2
Lesson Objective
Students will be able to:
Define the derivative of a function at a number a as a limit.
Apply the limit definition of the derivative to find f'(x) for polynomial and simple rational functions.
Relate the value of the derivative at a point to the slope of the tangent line at that point.
Translate between the notation for a function and its derivative.
How can we find the slope of a curve at a single, exact point?
What happens to the secant line as the distance between two points (h) approaches zero?
Why is the derivative defined as a limit?
How does the "Difference Quotient" transform into the derivative function f'(x)?
Derivative
Instantaneous Rate of Change
Tangent Line
Limit Definition of the Derivative
Difference Quotient
Differentiability
Leibniz Notation
Power Rule (Preview/Conceptual)
F-IF.7: Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.
F-IF.8: Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
F-IF.9: Compare properties of two functions each represented in a different way
This lesson introduces the Limit Definition of the Derivative. Students will visualize a secant line where one point is fixed at x and the second point is x+h. By applying a limit as h approaches 0, students "slide" the second point toward the first until the secant line becomes the tangent line. The lesson involves heavy algebraic manipulation (rationalizing, expanding binomials, and simplifying fractions) to evaluate the limit and find the general derivative formula f'(x).
Purpose: To move students away from "estimating" change to "calculating" exact change. This provides the mathematical justification for all the "shortcuts" (like the Power Rule) they will learn later.
DOK Level 1 (Recall): State the formal definition of the derivative using limit notation.
DOK Level 2 (Skill/Concept): Use the limit definition to find the derivative of a linear or quadratic function.
DOK Level 3 (Strategic Thinking): Recognize a given limit expression as the derivative of a specific function at a specific point (a common AP multiple-choice task).
DOK Level 4 (Extended Thinking): Justify why a derivative does not exist at a sharp turn or vertical tangent using one-sided limits.
For Struggling Learners (Scaffolding):Algebraic Checkpoints: Desmos Visualization: Use a pre-made Desmos slider where students can manually shrink h to see the secant line become the tangent line.
For Advanced Learners (Extension):
Non-Polynomials: Have them attempt the limit definition to practice rationalizing and common denominators.
AP College Board Assessments