Lesson 7: 2-7 Derivative of cos x, sin x, e^x and ln x
Duration of Days: 2
Lesson Objective
Students will be able to:
Understand and apply the derivative rules for cos(x), sin(x), e^x, and ln(x).
Combine these rules with other differentiation rules to find derivatives of more complex functions.
Solve problems involving rates of change, related rates, and optimization problems that require the use of these derivative rules.
What are the derivatives of cos(x), sin(x), e
x, and ln(x)?
How can you derive these formulas using the limit definition of the derivative?
How do these derivatives relate to the graphs of the original functions?
Sine Function
Cosine Function
Periodic Function
Transcendental Function
Circular Functions
Oscillation
Trigonometric Identity
Squeeze Theorem
Phase Shift
Amplitude
Radian Measure
MA.2.B.1: Define the derivative of a function at a point as the limit of difference quotients and interpret the derivative as an instantaneous rate of change.
MA.2.B.2: Apply the definition of the derivative to calculate derivatives of polynomial, rational, and trigonometric functions.
MA.2.B.3: Use differentiation rules to find derivatives of algebraic, trigonometric, exponential, and logarithmic functions, including the use of the chain rule, product rule, and quotient rule.
MA.2.B.4: Interpret the meaning of the derivative in various contexts, including velocity, acceleration, rates of change, and optimization problems.
This lesson introduces the "circular" nature of trigonometric derivatives. Students explore how the rate of change of a sine wave creates a cosine wave, and how the rate of change of a cosine wave creates a "flipped" (negative) sine wave. The lesson typically includes a brief look at the limit-based proof but focuses heavily on procedural fluency and applying these new rules alongside the power and constant multiple rules.
Purpose: To expand the student's ability to model periodic motion (like sound waves or pendulums). It moves the course beyond algebraic functions and into transcendental functions.
DOK Level 1 (Recall): Memorizing
DOK Level 2 (Skill/Concept): Differentiating a combination function.
DOK Level 3 (Strategic Thinking): Finding the equation of a tangent line to a trigonometric curve at a specific point.
For Struggling Learners (Scaffolding):The "Trig Clock": Create a visual circle showing to help students remember the cycle of derivatives.
For Advanced Learners (Extension):Higher-Order Derivatives: Ask students to find the 4th or 8th derivative of sin x and identify the repeating pattern.
AP College Board Classroom Assessments