Chapter 2: Derivatives

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Course

MATH161 (Calculus I)

Overview

This chapter covers Derivatives from Stewart Calculus 9th Edition.

Each section contains:

Skill breakdowns with learning objectives
Practice problems at multiple difficulty levels
Links to related skills and prerequisites

Sections

Section 2.1: Derivatives and Rates of Change

Skill	Description	Difficulty
Finding Tangent Lines Using the Derivative	Derivatives	Intermediate
Instantaneous Velocity from the Derivative	Derivatives	Intermediate
Interpreting Rates of Change with Units	Derivatives	Intermediate
Rates of Change	Derivatives	Intermediate
Tangent Lines	Derivatives	Intermediate
The Derivative Definition	Derivatives	Intermediate
The Derivative at a Point	Derivatives	Intermediate

Section 2.2: The Derivative as a Function

Skill	Description	Difficulty
Differentiability Implies Continuity	Derivatives	Intermediate
Higher Derivatives	Derivatives	Intermediate
Other Notations	Derivatives	Beginner
Sketching f’ from f	Derivatives	Intermediate
The Derivative Function	Derivatives	Intermediate
When Is a Function Differentiable?	Derivatives	Intermediate

Section 2.3: Differentiation Formulas

Skill	Description	Difficulty
Derivative of Constants	Derivatives	Beginner
General Power Functions	Derivatives	Intermediate
Sum and Difference Rules	Derivatives	Beginner
Sum, Difference, and Constant Multiple Rules	Differentiation Formulas	Beginner
The Constant Multiple Rule	Derivatives	Beginner
The Power Rule	Differentiation Formulas	Beginner
The Product Rule	Differentiation Formulas	Intermediate
The Quotient Rule	Differentiation Formulas	Intermediate

Section 2.4: Derivatives of Trigonometric Functions

Skill	Description	Difficulty
Derivatives of Other Trig Functions	Derivatives	Intermediate
Derivatives of Other Trigonometric Functions	Trigonometric Derivatives	Intermediate
Derivatives of Sine and Cosine	Trigonometric Derivatives	Intermediate
Differentiating Trig Expressions	Derivatives	Intermediate
Differentiating Trigonometric Expressions	Derivatives	Intermediate
Special Trigonometric Limits	Limits	Intermediate
Trigonometric Derivative Formulas	Derivatives	Intermediate

Section 2.5: The Chain Rule

Skill	Description	Difficulty
Applying the Chain Rule	Derivatives	Intermediate
Generalized Power Rule	Derivatives	Intermediate
Identifying Composite Functions	Derivatives	Intermediate
Nested Chain Rule	Differentiation Rules	Advanced
Recognizing Composite Functions	Derivatives	Beginner
The Chain Rule	Differentiation Rules	Intermediate

Section 2.6: Implicit Differentiation

Skill	Description	Difficulty
Implicit Differentiation	Derivatives	Intermediate
Implicitly Defined Functions	Derivatives	Intermediate
Second Derivatives of Implicit Functions	Derivatives	Advanced
Tangent Lines to Implicit Curves	Derivatives	Intermediate

Skill	Description	Difficulty
Average vs. Instantaneous Rate of Change	Derivatives and Rates of Change	Intermediate
Interpreting Derivatives in Context	Derivatives and Rates of Change	Intermediate
Physics Applications	Derivatives	Intermediate
Rates in Natural Sciences	Derivatives	Intermediate
Rectilinear Motion	Derivatives and Rates of Change	Intermediate

Skill	Description	Difficulty
Geometric Related Rates	Derivatives	Advanced
Implicit Differentiation with Respect to Time	Applications of Derivatives	Intermediate
Related Rates Problem Solving	Applications of Derivatives	Advanced
Setting Up Related Rates Problems	Applications of Derivatives	Intermediate
Solving Related Rates Problems	Derivatives	Advanced

Section 2.9: Linear Approximations and Differentials

Skill	Description	Difficulty
Differentials	Derivatives	Intermediate
Differentials	Derivatives	Intermediate
Error Estimation with Differentials	Derivatives	Intermediate
Error Estimation with Differentials	Derivatives	Intermediate
Linearization	Derivatives	Intermediate

Prerequisites

Before starting Chapter 2, review relevant prerequisite skills from earlier chapters.

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