Using derivatives to describe the rate of change of one variable with respect to another variable allows students to understand change in a variety of contexts. In AP Calculus, students build the derivative using the concept of limits and use the derivative primarily to compute the instantaneous rate of change of a function. Applications of the derivative include finding the slope of a tangent line to a graph at a point, analyzing the graph of a function (for example, determining whether a function is increasing or decreasing and finding concavity and extreme values), and solving problems involving rectilinear motion. Students should be able to use different definitions of the derivative, estimate derivatives from tables and graphs, and apply various derivative rules and properties. In addition, students should be able to solve separable differential equations, understand and be able to apply the Mean Value Theorem, and be familiar with a variety of real-world applications, including related rates, optimization, and growth and decay models.

Click on a link to jump ahead to the following sections:

- Unit 1: Introduction to Derivatives
- 2.10 Ad-Out - Instantaneous Velocity
- 2.11 That’s So Derivative - Introduction to Differential Calculus
- 2.12 The Derivative as a Function - Multiple Representations
- 2.13 Get Real - Interpreting, Estimating and Using the Derivative
- Video - Derivatives... How? - Video from Nancy Pi

- 2.14 Let Me Take You Higher - The Second Derivative
- 2.15 MakingConnections - Limits, Continuity and Differentiability
- 2.16 On a Tangent - The Tangent Line Approximation
- Video - How to Find the Equation of a Tangent Line with Derivatives - Video from Nancy Pi

- Unit 2: The Basics of Differentiation
- 2.21 Power Up! - Computing Basic Differentiation
- 2.22 Periodic Behavior - Differentiating Basic Trig Functions
- 2.23 Combining Forces - The Product and Quotient Rules of Differentiation
- 2.24 Tan-z and the Reciprocals - Derivatives of Other Trigonometric Functions
- 2.25 The Structure of Compositions - The Chain Rule
- 2.26 Anything You Can Do… I Can Un-Do - Derivatives of Inverse Functions/li>
- 2.27 Buried Treasures - Derivatives of Functions Given Implicitly
- 2.2H L’Hôpital to the Rescue - Using Derivatives to Evaluate Limits

- Unit 3: Applying derivatives to analyze functions
- 2.31 Mind Your Behavior - Using Derivatives to Identify Extreme Values
- 2.32 To Be or Not To Be - Mean Value Theorem, and the other Existence Theorems
- 2.34 Max Effort with the EVT - Global Optimization
- 2.35 Get Real - Applied Optimization Problems
- 2.36 Related Rates - complete lesson on Related Rates

- Assessment Tasks
- FRQ - Related Rates - from AP Central
- Project - Cooperative Task - Related Rates
- Summative Assessment - Group or Individual Task

The following are links to resources for Derivatives:

- Differentiation: definition and basic derivative rules - the introductory unit on Khan Academy
- Differentiation: composite, implicit, and inverse functions - the next unit on Khan Academy
- Differentiation: definition and basic derivative rules - the next unit on Khan Academy
- Applying derivatives to analyze functions - the final unit on Khan Academy

The following are other free resources:

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