Chapter 3.2

The Derivative as a Function

• If f ’ exists at a particular x then f is differentiable (has a derivative) at x

• Differentiation is the process of calculating a derivative

Derivatives from Definition

1)(

x

xxf Find f ’

h

xfhxfxf

h

)()(lim)('

0

hxx

hxhx

h

11)(lim

0

h

xhxhxx

xhxxhx

h

)1](1)[(]1)[(

)1](1)[()1)((

lim0

)1](1)[(

]}1)[({)1)((lim

0

xhxh

hxxxhxh )1](1)[(

lim22

0

xhxh

xxhxhxhxxh

)1](1)[(lim

0

xhxh

hh )1](1)[(

1lim

0

xhxh 2)1(

1

x

Derivatives from Definition

xxf )( Find f ’

h

xfhxfxf

h

)()(lim)('

0

h

xhxxf

h

0lim)('

xhx

xhx

h

xhxh 0lim

)(lim

0 xhxh

xhxh

)(lim

0 xhxh

hh

)(

1lim

0 xhxh

xxx 2

1

)(

1

0x

Tangent Line• Last example, the slope of the curve at x = 4 is

• The tangent is the line through the point (4,2) with slope 1/4

4

1

42

1)4(' f

)4(4

12 xy 1

4

1 xy

Derivative Notations

Derivative Values at a specific number x = a

Graphing Derivatives

• Estimating the slopes of the graph by plotting the points (x, f ’(x))

• Connect the points to make the curve y = f ’(x)

• What the graph tells us– Where the rate of change of f is

positive, negative or zero– The rough size of the growth rate at

any x and its size in relations to the size of f(x)

– Where the rate of change itself is increasing or decreasing

Interval and One-Sided Derivatives

• Differentiable on an interval– Derivative at each point on the interval– Differentiable on a closed interval [a,b] if it is

differentiable at the interior (a,b) and if the right-hand and left-hand derivatives exist at the end points a and b respectively, that is

aat derivative hand-right ,)()(

lim0 h

afhafh

bat derivative hand-left ,)()(

lim0 h

bfhbfh

Interval and One-Sided Derivatives

• Examples:

Function NOT Have a Derivative at a Point

Differentiable Functions

• A function is continuous at every point where it has a derivative