Lesson 22.2

The Limit Definition of the Derivative

Mastering the fundamental difference quotient.

Introduction

The derivative of a function at a point is defined as a limit. Thisdifference quotient formula is the foundation of all derivative rules you'll learn in calculus.

1

Prerequisite Connection

You understand limits and tangent line slopes.

2

Today's Increment

We apply the limit definition to find derivatives of basic functions.

3

Why This Matters

Understanding the definition helps you understand WHY derivative rules work.

Key Concepts

The Limit Definition

Alternative Form

Notation

All of these mean the same thing:

Worked Examples

Example 1: Linear Function (Basic)

Find for

(constant slope!)

Example 2: Quadratic Function (Intermediate)

Find for

Example 3: Square Root (Advanced)

Find for

Rationalize:

Common Pitfalls

Algebra errors in f(x+h)

, NOT

Dividing by h before simplifying

You can only cancel h after factoring it out of the numerator.

Real-World Application

Velocity and Acceleration

If s(t) is position, then s'(t) is velocity and s''(t) is acceleration. The limit definition shows exactly how we compute these from position data.

Practice Quiz

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