The Derivative Function

Objective: To define and use the derivative function

Definition 2.2.1

• The function defined by the formula

• is called the derivative of f with respect to x. The domain of consists of all x in the domain of for which the limit exists.

• Remember, this is called the difference quotient.

h

xfhxfxf

h

)()(lim)(

0

/

/f

/f f

Example 1

• Find the derivative with respect to x of and use it to find the equation of the tangent line to

at

• Note: The independent variable is x. This is very important to state. Later, we will be taking derivatives with respect to other independent variables.

1)( 2 xxf

1)( 2 xxf .2x

Example 1

• Find the derivative with respect to x of and use it to find the equation of the tangent line to at

1)( 2 xxf

1)( 2 xxf .2x

h

xhx

h

xfhxfxf

h

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

22

0

/

Example 1

• Find the derivative with respect to x of and use it to find the equation of the tangent line to at

1)( 2 xxf

1)( 2 xxf .2x

h

xhx

h

xfhxfxf

h

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

22

0

/

xhxh

hxh

h

xhxhxxf

h22

)2(112lim)(

222

0

/

Example 1

• The slope of the tangent line to at is When , so the equation of

the tangent line at is

12 xy

.4)2(/ f2x

5,2 yx2x

34

)2(45

xy

or

xy

Example 1

• We can also use the other formula to find the derivative of . 1)( 2 xxf

01

01 )()(lim

01 xx

xfxfxx

01

20

21

01

20

21 ]1[]1[

lim01 xx

xx

xx

xxxx

00101

0101 2))((

xxxxx

xxxx

Example 2

a) Find the derivative with respect to x of xxxf 3)(

h

xxhxhx

h

xfhxfxf

h

][)]()[()()(lim)(

33

0

/

Example 2

a) Find the derivative with respect to x of xxxf 3)(

h

xxhxhx

h

xfhxfxf

h

][)]()[()()(lim)(

33

0

/

h

xxhxhxhhxxh

][]33[lim

33223

0

Example 2

a) Find the derivative with respect to x of xxxf 3)(

h

xxhxhx

h

xfhxfxf

h

][)]()[()()(lim)(

33

0

/

h

xxhxhxhhxxh

][]33[lim

33223

0

13)133(33

lim 222322

0

xh

hxhxh

h

hhxhhxh

Example 2

• We can use the other formula to find the derivative of .xxxf 3)(

01

01 )()(lim

01 xx

xfxfxx

01

0130

31

01

0301

31 ][][][][

lim01 xx

xxxx

xx

xxxxxx

01

2001

2101

01

012001

2101 ]1))[(()())((

lim01 xx

xxxxxx

xx

xxxxxxxxxx

13)1(lim 20

2010

21

01

xxxxxxx

Example 2

• Lets look at the two graphs together and discuss the relationship between them.

Example 2

• Since can be interpreted as the slope of the tangent line to the graph at it follows that

is positive where the tangent line has positive slope, is negative where the tangent line has negative slope, and zero where the tangent line is horizontal.

)(/ xf

)(xfy

)(/ xf

x

Example 3

• At each value of x, the tangent line to a line is the line itself, and hence all tangent lines have slope m. This is confirmed by:

h

bmxbhxm

h

xfhxfh

][)()()(lim

0

mh

mh

h

bmxbmhmxh

0lim

Example 4

• Find the derivative with respect to x of

• Recall from example 4, section 2.1 we found the slope of the tangent line of was , thus,

• Memorize this!!!!

xxf )(

xy x2

1

xxf

2

1)(/

0000

0

0

0 1

))((lim

0 xxxxxx

xx

xx

xxxx

Example 4

• Find the derivative with respect to x of • Find the slope of the tangent line to at x = 9.

• The slope of the tangent line at x = 9 is

xxf )(

xxf )(

6

1

92

1)9(/ f

Example 4

• Find the derivative with respect to x of • Find the slope of the tangent line to at x = 9.• Find the limits of as and as and explain what those limits say about the graph of

xxf )(

xxf )(

)(/ xf 0x x

.f

Example 4

• Find the limits of as and as and explain what the limits say about the graph of • The graphs of f(x) and f /(x) are shown. Observe that if , which means that all tangent lines

to the graph of have positive slopes, meaning that the graph becomes more and more vertical as

and more and more horizontal as

)(/ xf 0x

.f

0)(/ xf 0x

xxf )(

xy

0x .x

Instantaneous Velocity

• We saw in section 2.1 that instantaneous velocity was defined as

• Since the right side of this equation is also the definition of the derivative, we can say

• This is called the instantaneous velocity function, or just the velocity function of the particle.

h

tfhtfv

hinst

)()(lim

0

h

tfhtftftv

h

)()(lim)()(

0

/

Example 5

• Recall the particle from Ex 5 of section 2.1 with position function . Here f(t) is measured in meters and t is measured in seconds. Find the velocity function of the particle.

2251)( tttfs

Example 5

• Recall the particle from Ex 5 of section 2.1 with position function . Here f(t) is measured in meters and t is measured in seconds. Find the velocity function of the particle.

2251)( tttfs

h

tththt

h

tfhtftv

hh

]251[])(2)(51[lim

)()(lim)(

22

00

h

hhth

h

hththth

5245]2[2lim

2222

0

thth

45)524(lim0

Differentiability

• Definition 2.2.2 A function is said to be differentiable at x0 if the limit

exists. If f is differentiable at each point in the open interval (a, b) , then we say that is differentiable on (a, b), and similarly for open intervals of the form . .In the last case, we say that it is differentiable everywhere.

h

xfhxfxf

h

)()(lim)( 00

00

/

),(),(),,( andba

Differentiability

• Definition 2.2.2 A function is said to be differentiable at x0 if the limit

exists. When they ask you if a function is differentiable on the AP Exam, this is what they want you to reference.

h

xfhxfxf

h

)()(lim)( 00

00

/

Differentiability

• Geometrically, a function f is differentiable at x if the graph of f has a tangent line at x. There are two cases we will look at where a function is non-differentiable.

1. Corner points2. Points of vertical tangency

Corner points

• At a corner point, the slopes of the secant lines have different limits from the left and from the right, and hence the two-sided limit that defines the derivative does not exist.

Vertical tangents

• We know that the slope of a vertical line is undefined, so the derivative makes no sense at a place with a vertical tangent, since it is defined as the slope of the line.

Differentiability and Continuity

• Theorem 2.2.3 If a function f is differentiable at x, then f is continuous at x.

• The inverse of this is not true. If it is continuous, that does not mean it is differentiable (corner points, vertical tangents).

Differentiability and Continuity

• Theorem 2.2.3 If a function f is differentiable at x, then f is continuous at x.

• Since the conditional statement is true, so is the contrapositive:

• If a function is not continuous at x, then it is not differentiable at x.

Other Derivative notations

• We can express the derivative in many different ways.

• Please note that these expressions all mean the derivative of y with respect to x.

// )]([)( ydx

dyxf

dx

dxf

Other formulas to use

• There are several different formulas you can use to find the derivative of a function. The only ones we will use are:

h

xfhxfxf

h

)()(lim)(

0

/

xw

xfwfxf

xw

)()(lim)(/

0

0/ )()(lim)(

0 xx

xfxfxf

xx

Homework

• Section 2.2• Page 152-153• 1-25 odd, 31• For numbers 15,17,19, use formula 13, not formula

12.

Example

• Evaluate:

h

xxhxhxh

6346)(3)(4lim

2323

0

Example

• Evaluate:

• This is the definition of the derivative of .

• The answer is . You are not supposed to do any work, just recognize this!

h

xxhxhxh

6346)(3)(4lim

2323

0

634 23 xx

xx 612 2