Lesson 1: 3-1 The Chain Rule
Duration of Days: 3
Lesson Objective
Students will be able to understand that the chain rule is used to differentiate composite functions, i.e., functions that are composed of two or more simpler functions.
Recognize the structure of composite functions and identify the "inner" and "outer" functions in a composition.
What is the chain rule, and why is it necessary for differentiating composite functions?
How can you identify the "outer" and "inner" functions in a composite function?
How does the chain rule relate to the concept of function composition?
Composite Function
Inner Function
Outer Function
Decomposition
The "General Power Rule"
Leibniz Notation
Function Composition
Derivative of the "Blob" (
Nested Functions
Chain of Rates
F-BF: Building Functions
F-IF.8: Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
This lesson introduces the "Inside-Outside" rule. Students learn to decompose a composite function f(g(x)) into its components. The lesson emphasizes the Leibniz notation to show how rates of change multiply. Practice transitions from simple power functions with binomial bases to more complex trigonometric compositions.
Purpose: To move beyond basic differentiation and enable students to find the rate of change for complex, real-world models where one variable depends on another, which in turn depends on a third.
DOK 1: Recall and Reproduction
Tasks involve recognizing and recalling the Chain Rule formula.
Students practice differentiating simple composite functions using the Chain Rule.
DOK 2: Skills and Concepts
Tasks require students to apply the Chain Rule to more complex functions, including those involving trigonometric, logarithmic, and exponential functions.
Students may need to combine the Chain Rule with other differentiation rules
DOK 3: Strategic Thinking and Reasoning
Tasks involve analyzing functions and determining the most appropriate approach to differentiation.
Students may need to break down complex functions into simpler components to apply the Chain Rule effectively.
Differentiation Strategies
For Struggling Learners: Use the "U-Substitution" visual method. Have students literally rewrite the inner function as u and find u separately before plugging them back into the "shell" derivative.
For Advanced Learners: Introduce "Triple Composites" , requiring the Chain Rule to be applied twice in a single problem.
AP College Board Classroom Assessments