Lesson 6: 3-6 Calculating Higher-Order Derivatives

What is a higher-order derivative?
How does it relate to the original function?
What is the notation for higher-order derivatives?
How can you interpret the meaning of higher-order derivatives in real-world contexts?
How do the rules for finding derivatives apply to higher-order derivatives?

First Derivative
Second Derivative
Higher-Order Derivatives
Derivative
Iterative Differentiation
Acceleration (Physical Interpretation)
Concavity (Graphical Interpretation)
Substitution (in Implicit Second Derivatives)

C-ID.1: Derive the derivative of a function from its graph at a point.
C-ID.2: Define the derivative of a function at a point as the limit of a difference quotient.
C-ID.3: Interpret the derivative as a rate of change.
C-ID.4: Determine whether a function is differentiable at a point.

This lesson introduces the concept of differentiating a derivative. Students practice finding f''(x), f'''(x). The core challenge of this lesson is Algebraic Stamina. Students will apply the Power, Product, and Chain rules repeatedly. A significant portion of the lesson is dedicated to Implicit Second Derivatives, where students must differentiate and substitute the original first derivative back into the equation to get an answer in terms of x and y only.
Purpose: To prepare for Unit 4 (Contextual Applications), where the second derivative represents acceleration in physics and concavity in curve sketching.

DOK 1: Recall and Reproduction

Tasks involving direct application of differentiation rules to find the second or third derivative of a simple function.
DOK 2: Skills and Concepts

Tasks requiring the use of multiple differentiation rules (e.g., product rule, quotient rule, chain rule) to find higher-order derivatives.
Problems involving implicit differentiation to find higher-order derivatives.
DOK 3: Strategic Thinking

Tasks that involve analyzing the behavior of a function using higher-order derivatives, such as finding intervals of concavity and inflection points.
Problems requiring the use of higher-order derivatives to solve optimization or related rates problems.
DOK 4: Extended Thinking

Tasks that involve creating models or solving non-routine problems using higher-order derivatives.
Problems that require the use of higher-order derivatives to analyze complex functions or physical phenomena.

For Struggling Learners: Use a "Derivative Ladder" visual. Have them write each successive derivative on a new rung to keep the work organized and avoid skipping steps.
For Advanced Learners: Ask them to find a general formula for the n^th derivative of f(x) = 1/x. This requires them to use inductive reasoning to find a pattern involving factorials and alternating signs.

AP College Board Classroom Assessments