Lesson 5: 3-5 Selecting Procedures for Calculating Derivatives
Duration of Days: 2
Lesson Objective
Student Learning Objectives
Upon completion of this lesson, students will be able to:
Identify and Apply Formulas: Recall and apply the derivative formulas for arcsin u, arccos u, and arctan u.
Execute the Chain Rule: Correctly differentiate expressions where the argument of the inverse trig function is itself a function.
Connect to Geometry: Explain the presence of the square root and x^2 in the formulas using right-triangle relationships.
Simplify Resulting Expressions: Clean up complex fractions that arise when u is a fraction or a radical.
What is the power rule, and when is it used?
What is the product rule, and how is it applied to functions that are the product of two simpler functions?
What is the quotient rule, and how is it used to differentiate functions that are the quotient of two simpler functions?
What is the chain rule, and how is it used to differentiate composite functions?
How can I identify when to use implicit differentiation?
How can I find the derivative of an inverse function?
Inverse Sine
Inverse Cosine
Inverse Tangent
Restricted Domain
Radical Expression
Pythagorean Identity Substitution
Tangent Line Approximation
C-ID.A.2: Derive the derivative of a function from its graph at a point.
C-ID.A.3: 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.
This lesson introduces the derivative formulas for the six inverse trigonometric functions, with a heavy emphasis on arcsin x, arccos x, and arctan x (the three most common on the AP Exam). Students explore how these derivatives are derived using implicit differentiation and right-triangle trigonometry. The lesson moves into applying the Chain Rule.
Purpose: To allow students to differentiate functions involving angles and rotations. These derivatives are also essential "backwards" tools for Unit 6 (Integration), where students must recognize 1/1+x^2 as the integral of arctan x.
DOK 1: Recall and Reproduction
Tasks involve recognizing and recalling differentiation rules (e.g., power rule, product rule, quotient rule, chain rule).
Students are asked to apply these rules to straightforward
DOK 2: Skills/Concepts
Tasks require students to apply multiple differentiation rules to more complex functions.
Students may need to simplify expressions or combine different rules within a single problem.
Students may be asked to analyze a function and determine the most efficient approach to find its derivative.
DOK 3: Strategic Thinking
Tasks require students to solve non-routine problems that involve selecting and applying appropriate differentiation techniques.
Students may need to break down complex functions into simpler components or use multiple approaches to find a solution.
Students may be asked to justify their choice of differentiation rules and explain their reasoning.
DOK 4: Extended Thinking
Tasks require students to analyze complex functions and develop innovative approaches to find their derivatives.
Students may need to create their own differentiation rules or modify existing ones to solve unique problems.
Students may be asked to investigate the properties of functions and their derivatives, and make generalizations based on their findings.
For Struggling Learners: Focus strictly on the "Big Three" first. Point out that the "co" versions are simply the negative versions of their counterparts.
For Advanced Learners: Have students use a right triangle to prove why. This reinforces the Pythagorean identity in a calculus context.
AP College Board Classroom Assessments