Lesson 3: 2-3 Estimating Derivatives
Duration of Days: 1
Lesson Objective
Students will be able:
Apply the Power Rule to differentiate functions with positive, negative, and rational exponents.
Utilize the Constant Multiple and Sum/Difference rules to find the derivative of complex polynomial expressions.
Rewrite functions involving radicals or variables in the denominator into a form compatible with the Power Rule.
Determine the slope of a curve at a specific x-value using differentiation rules.
Write the full equation of a tangent line using the point-slope formula.
Is there a faster way to find the derivative of a polynomial without using the limit definition?
How does the derivative of a constant function relate to its graph?
How do the rules of exponents (negative and fractional) allow us to use the Power Rule on radicals and rational functions?
What is the relationship between the derivative of a sum of functions and the derivatives of the individual functions?
Constant Rule
Power Rule
Sum and Difference Rules
Constant Multiple Rule
Polynomial
Radicand
Rational Exponent
Negative Exponent
Normal Line (Perpendicular to the tangent)
Point-Slope Form
C-D.1: Define the derivative of a function at a point as the limit of the difference quotients. Interpret the derivative as the instantaneous rate of change
Lesson Description
This lesson introduces the fundamental shortcuts of differentiation. Students learn the Constant Rule, the Power Rule, the Constant Multiple Rule, and the Sum and Difference Rules. A major focus of this lesson is "algebraic preparation"—rewriting radicals as fractional exponents and moving variables from the denominator to the numerator so the Power Rule can be applied.
Purpose: To build computational fluency. These rules are the "arithmetic" of calculus; students must be able to perform them automatically to tackle the complex applications later in the course.
DOK Level 1 (Recall): Applying the Power Rule to simple polynomials.
DOK Level 2 (Skill/Concept): Rewriting complex terms (radicals or fractions) into power form before differentiating.
DOK Level 3 (Strategic Thinking): Finding the equation of a tangent line at a specific point by using the derivative rules to find the slope.
For Struggling Learners (Scaffolding):
The "Bring it Down, Subtract One" Mantra: Use physical gestures or rhythmic phrases to reinforce the two-step Power Rule process.
For Advanced Learners (Extension):
Higher-Order Derivatives: Introduce the second and third derivatives (f''(x), f'''(x)) and ask them to find where the "acceleration" of a polynomial becomes zero.
AP College Board Assessments