Lesson 22.2

The Limit Definition of the Derivative

Mastering the fundamental difference quotient.

Introduction

Mastering the fundamental difference quotient.

You understand limits and tangent line slopes.

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

Understanding the definition helps you understand WHY derivative rules work.

The Limit Definition

Alternative Form

Notation

All of these mean the same thing:

Find for

(constant slope!)

Find for

Find for

Rationalize:

Algebra errors in f(x+h)

, NOT

Dividing by h before simplifying

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

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.