5128_Ch03_pp098-184

Chapter 3 Overview
Section 3.1 Derivative of a Function 99
In Chapter 2, we learned how to find the slope of a tangent to a curve as the limit of the
slopes of secant lines. In Example 4 of Section 2.4, we derived a formula for the slope of
the tangent at an arbitrary point a, 1a on the graph of the function f x 1x and
showed that it was 1a 2 .
This seemingly unimportant result is more powerful than it might appear at first glance,
as it gives us a simple way to calculate the instantaneous rate of change of f at any point.
The study of rates of change of functions is called differential calculus, and the formula
1a 2 was our first look at a derivative. The derivative was the 17th-century breakthrough
that enabled mathematicians to unlock the secrets of planetary motion and gravitational
attraction—of objects changing position over time. We will learn many uses for derivatives
in Chapter 4, but first we will concentrate in this chapter on understanding what derivatives
are and how they work.
3.1
What you’ll learn about
• Definition of Derivative
• Notation
• Relationships between the
Graphs of f and f
• Graphing the Derivative from
Data
• One-sided Derivatives
. . . and why
The derivative gives the value of
the slope of the tangent line to a
curve at a point.
Derivative of a Function
Definition of Derivative
In Section 2.4, we defined the slope of a curve y f x at the point where x a to be
m lim f a h f a
.
h→0 h
When it exists, this limit is called the derivative of f at a. In this section, we investigate
the derivative as a function derived from f by considering the limit at each point of the domain
of f.
DEFINITION Derivative
The derivative of the function f with respect to the variable x is the function f
whose value at x is
f x lim f x h f x
, (1)
h→0 h
provided the limit exists.
The domain of f , the set of points in the domain of f for which the limit exists, may be
smaller than the domain of f. If f x exists, we say that f has a derivative (is differentiable)
at x. A function that is differentiable at every point of its domain is a differentiable
function.
EXAMPLE 1
Applying the Definition
Differentiate (that is, find the derivative of) f x x 3 .
continued