2.3 Introduction to Finding Derivatives

In this lesson, you will learn that the derivative is the limit of the slopes of the secant lines and that gives the slope of the tangent line. You will learn that the derivative is also the instantaneous rate of change of a function at x = a. Where the derivative exists, the function is called differentiable. Furthermore, the derivative itself is a function of x.

Lesson Objectives

Upon completion of the lesson 2.3, you will be able to:

• • Visualize the tangent line to the graph of a function.

  • Apply limits to find the slope of the tangent line to a curve at x=a or the instantaneous rate of change of a function at x = a or to find the derivative of a function at x=a.

  • Recognize that finding the slope of the tangent line to a curve at x=a, or finding the instantaneous rate of change of a function at x=a or finding the derivative of a function at x=a are synonymous.

  • Recognize and utilize at least 3 different symbolic notations for the derivative.

  • Estimate the graph of the derivative of a function from the graph of the function and distinguish between the graph of a function and its derivative function.

  • Identify where (at what x-values) the function is continuous, differentiable, not continuous, or not differentiable.

Lesson Content

View all of the following instructional videos. These will help you master the objectives for this module.

• 1. YouTube video: The Definition of the Derivative

• 1. The Derivative [University of Houston] (Slope of the tangent line, definition of the derivative, differentiability and non-differentiability at a point)

  2. YouTube video: Finding the Slope of a Tangent Line to a Curve

• 1. YouTube video: More on Finding the Slope of a Tangent Line to a Curve

1. Tangents and Derivatives at a Point [UMass Lowel]

2. The Derivative as a Function [UMass Lowell]

Lesson Readings

The following required readings cover the content for this module. As you go through each reading, pay close attention to the content that will help you learn the objectives for this module.

• 1. The Definition of the Derivative [Paul's Online Math Notes]

  2. Graph of Derivative [Auburn University] (two ways of interpret derivative, relating graph of function to graph of derivative, where the derivative is undefined)

Lesson Practice Exercises/Activities

Make your way through each of the practice exercises. This is where you will take what you have learned from the lesson content and lesson readings and apply it by solving practice problems.

• 1. The Definition of the Derivative [Paul's Online Math Notes]

  2. The Derivative – Intuitive [Khan Academy]

  3. Note: This site requires Chrome, Firefox, or Safari.

  4. Derivatives 1 [Khan Academy]

  5. Note: This site requires Chrome, Firefox, or Safari.

Additional Resources

Below are additional resources that help reinforce the content for this module.

• 1. Tangent Lines and the Definition of the Derivative [Eastern Iowa Community Colleges]

  2. (slope of the secant line and the definition of the derivative, differentiable on an interval– one-sided derivatives, when does a function not have a derivative at a point?)

  3. Secant Lines, Tangent Lines, and Limit Definition of a Derivative [UC Santa Barbara]

  4. - Brief Review only