Lesson 6: 2-6 Derivative Rule: Constant, Sum, Difference, ad Constant Multiple
Duration of Days: 1
Lesson Objective
Students will be able to:
Apply the Constant Rule to identify that the derivative of any constant is zero.
Apply the Constant Multiple Rule to differentiate terms with coefficients.
Use the Sum and Difference Rules to differentiate polynomials with multiple terms.
Combine all basic rules to find the derivative of a general polynomial function.
Solve for the slope of a tangent line using these combined rules.
What is the derivative of a constant function?
Can you give an example of a constant function and its derivative?
How do you find the derivative of a function that is the sum or difference of two or more functions?
Can you provide an example of a function that involves both addition and subtraction, and find its derivative?
How does the constant multiple rule affect the derivative of a function?
Can you give an example of a function multiplied by a constant, and find its derivative?
Constant Rule
Constant Multiple Rule
Sum Rule
Difference Rule
Linearity of the Derivative
Coefficient
Term
Horizontal Tangent
2.A: Identify common underlying structures in different mathematical representations.
2.D: Identify how mathematical characteristics or properties of functions are related in different representations.
This lesson formalizes the "arithmetic" of derivatives. Students learn that differentiation is a linear operator: the derivative of a sum is the sum of the derivatives. Students will practice differentiating multi-term polynomials by applying the Power Rule (from 2.5) to each term individually. A major focus is recognizing that constants (like k) stay attached to their variables during the process but "vanish" if they are standing alone.
Purpose: To move students from differentiating single terms to differentiating entire functions. This builds the foundation for solving "real" functions.
DOK Level 1 (Recall): Differentiating functions.
DOK Level 2 (Skill/Concept): Identifying that in a function a constant multiple and can be pulled out before differentiating.
DOK Level 3 (Strategic Thinking): Finding the x-coordinates where a polynomial has a horizontal tangent line (setting the derivative equal to zero and solving).
For Struggling Learners (Scaffolding):"The Eraser Test": For the Constant Rule, ask: "Is the function changing?" If f(x)=7, it’s a flat line. A flat line has a slope of zero.
Color-Coding Terms: Have students use different colors for each term in a polynomial to remind them to differentiate them separately.
For Advanced Learners (Extension):Conceptual Proof: Application: Ask students to find the derivative of a polynomial to see the general form of all quadratic derivatives.
AP College Board Classroom Assessments