Chapter 2: Differentiation

Overview

This chapter introduces the foundational concepts of differentiation in calculus, including the definition of the derivative, tangent lines, basic differentiation rules, rates of change, and advanced techniques such as the product, quotient, and chain rules. Applications to velocity, position, and related rates are also covered.

2.1: The Derivative and the Tangent Line Problem

Definition and Interpretation

• Derivative of a Function: The derivative of a function at a point measures the instantaneous rate of change of the function at that point. It is defined as the limit:

• Tangent Line: The tangent line to the graph of at has slope and its equation is:

• Other Notations for Derivatives: , , ,

Differentiability and Continuity

• A function is differentiable at if the derivative exists at .

• If is differentiable at , then is also continuous at (Theorem 2.1).

• Alternate limit form for the derivative at :

• One-sided derivatives must be equal for differentiability:

and

Examples and Applications

• Example: Find the slope of the tangent line to at .

• Example: Find the derivative of using the limit process.

• Example: Find the derivative of using the limit process.

• Example: Find an equation of a line tangent to and parallel to .

• Example: Describe the -values at which is differentiable (using provided graphs).

2.2: Basic Differentiation Rules and Rate of Change

Basic Differentiation Rules

Rates of Change

• Position Function: gives the position of an object at time .

• Average Velocity:

• Instantaneous Velocity:

• Displacement: Change in position over a time interval.

Examples and Applications

• Example: Find the derivative of , , , .

• Example: Find an equation of the tangent line to at .

• Example: Determine the points where , , has a horizontal tangent line.

• Example: Find the average rate of change of over and compare with instantaneous rates at endpoints.

• Example: For a free-falling object under gravity:

• Example: A ball is thrown straight down from a 220-foot building with initial velocity ft/s. Find velocity after 3 seconds and after falling 108 feet.

2.3: Product and Quotient Rules & Higher Derivatives

Product Rule

• When differentiating the product of two functions:

• Steps:

  1. Identify and .

  2. Differentiate , multiply by .

  3. Differentiate , multiply by .

  4. Add the two results.

Quotient Rule

• When differentiating the quotient of two functions:

• Steps:

  1. Identify numerator and denominator .

  2. Differentiate , multiply by .

  3. Differentiate , multiply by .

  4. Subtract the second product from the first.

  5. Divide by .

Higher-Order Derivatives

• The second derivative is the derivative of , representing the rate of change of the rate of change.

• Higher-order derivatives are denoted as for the th derivative.

Examples and Applications

• Example: Use the product rule to find the derivative of .

• Example: Use the quotient rule to find the derivative of .

• Example: Find the derivative of .

• Example: Find the derivative of .

Additional info:

• Subsequent sections (not shown in images) likely cover the Chain Rule, Implicit Differentiation, and Related Rates, which are standard topics in a Calculus 1 differentiation chapter.

• Practice problems and quizzes are provided for each section to reinforce understanding.