Lesson 10: 2-10 Derivatives of tanx, cot x, sec x, and csc x
Duration of Days: 2
Lesson Objective
Students will be able to:
Identify the function as one of the four trigonometric functions.
Apply the corresponding derivative formula directly.
Simplify the result if possible.
What is the general process for differentiating trigonometric functions using the chain rule?
How do the derivatives of trigonometric functions compare to the derivatives of their inverse functions?
What is the significance of applying the quotient rule when differentiating functions like tan?
Reciprocal Trigonometric Functions
Co-functions
Quotient Rule Verification
Secant Function Derivative
Cosecant Function Derivative
Tangent Function Derivative
Cotangent Function Derivative
Trigonometric Identity Substitution
Point of Tangency
F-TF.A.2: Use trigonometric identities to solve problems.
F-TF.A.3: Prove trigonometric identities.
C-ID.A.2: Define the derivative of a function at a point.
C-ID.A.4: Differentiate functions.
This lesson completes the set of basic trigonometric derivatives. Students previously learned sin x and cosx; in 2.10, they use the Quotient Rule to prove why the derivatives of the other four trig functions are what they are. The lesson moves from the algebraic derivation to fluency in application—finding derivatives of functions that combine these trig terms with polynomials (using Product and Quotient rules) and evaluating them at specific points (unit circle integration).
Purpose: To expand the student's "differentiation toolbox" so they can handle any trigonometric expression without needing to convert back to sine and cosine. This builds the prerequisite skills for Unit 3 (Chain Rule).
DOK Level 1 (Recall): Memorizing the four new derivative rules.
DOK Level 2 (Skill/Concept): Using the Quotient Rule to derive tan x or sec x.
DOK Level 3 (Strategic Thinking): Solving multi-step problems where students must find the equation of a normal line or a horizontal tangent for a complex trigonometric function.
For Struggling Learners (Scaffolding):The "CO" Rule Mnemonic: Remind students that the derivative of every "CO" function results in a negative sign.
Reference Sheet: Provide a "Trig Derivative Graphic Organizer" that pairs the functions (e.g., tan and sec go together; cot and csc go together).
For Advanced Learners (Extension):
Multiple Representations: Have students identify where the graph of f(x) = sec x has a horizontal tangent line and justify their answer using the derivative.
AP College Board Classroom Assessments