Lesson 5: 2-5 Applying The Power Rule
Duration of Days: 1
Lesson Objective
Students will be able to define the derivative of a function at a point as the limit of the difference quotient.
Interpret the derivative as the instantaneous rate of change of a function.
Connect the derivative to the geometric interpretation of the slope of the tangent line to a curve.
Find the derivatives of functions involving constant multiples, sums, differences, and products of powers of x.
Determine the equations of tangent lines to curves at specific points using derivatives.
What is the power rule for differentiation?
How do you apply the power rule to find the derivative of a function?
What are the different forms of the power rule?
Power Rule
Constant Rule
Constant Multiple Rule
Sum and Difference Rules
Rational Exponent
Negative Exponent
Coefficient
Polynomial
Derivative Notation
FUN-3: Understand the concept of a function and use function notation.
FUN-3.A: Analyze functions using different representations.
FUN-3.A.1: Interpret functions that arise in applications in terms of the context.
This lesson introduces the most fundamental shortcut in Calculus.
Students move from the "long way" (limit definition) to the "short way." A significant portion of this lesson is actually an Algebra 2 review, as students must be able to rewrite radicals as fractional exponents and move variables from the denominator to the numerator using negative exponents before the Power Rule can be applied.
Purpose: To build computational efficiency. The Power Rule is the "bread and butter" of differentiation; students must master it to handle the more complex applications and theorems later in the course.
DOK Level 1 (Recall): Differentiating simple polynomials.
DOK Level 2 (Skill/Concept): Rewriting and then differentiating.
DOK Level 3 (Strategic Thinking): Finding the equation of a tangent line to a curve at a specific point using the Power Rule to find the slope.
For Struggling Learners (Scaffolding):
"The Power Slide": Use a physical or visual cue showing the exponent "sliding down" to the front and then being "reduced by one."
For Advanced Learners (Extension):
Higher-Order Derivatives: Have students find the 1st, 2nd, and 3rd derivatives of a polynomial and discuss when the derivative eventually becomes zero.
AP College Board Classroom Assessments