Section 2.1 – The Derivative and the Tangent Line Problem

Secant Line

A line that passes through two points on a curve.

Tangent Line Most people believe that a tangent line only intersects a

curve once. For instance, the first time most students see a tangent line is with a circle:

Although this is true for circles, it is not true for every curve:

Every blue line intersects the pink curve only once.

Yet none are tangents.

The blue line intersects the

pink curve twice. Yet it is a

tangent.

Local LinearityIf a function has a tangent line at a point, it is at least locally linear. Tangent Line Exists.

Local LinearityIf a function has a tangent line at a point, it is at least locally linear. Tangent Line Does Not Exist.

The function is NOT smooth at this point.

Local LinearityIf a function has a tangent line at a point, it is at least locally linear. A Tangent Line Exists at

Every Point.The function is Always smooth.

Tangent LineAs two points of a secant line are brought together, a tangent line

is formed. The slope of which is the instantaneous rate of change

Tangent LineAs two points of a secant line are brought together, a tangent line

is formed. The slope of which is the instantaneous rate of change

Tangent LineAs two points of a secant line are brought together, a tangent line

is formed. The slope of which is the instantaneous rate of change

Tangent LineAs two points of a secant line are brought together, a tangent line is formed. The slope of which

is the instantaneous rate of change

Tangent LineAs two points of a secant line are brought together, a tangent line

is formed. The slope of which is the instantaneous rate of change

Tangent LineAs two points of a secant line are brought together, a tangent line

is formed. The slope of which is the instantaneous rate of change

Slope of a Tangent LineIn order to find a formula for the slope of a tangent line, first look

at the slope of a secant line that contains (x1,y1) and (x2,y2):

f x2 f x1 x2 x1

yx

m

y2 y1x2 x1

f x1 x f x1 x

f x1 x f x1 x1 x x1

(x2,y2)

(x1,y1)

Δx

In order to find the slope of the tangent line, the change in x needs to be as small as

possible.

Instantaneous Rate of ChangeIf f is defined on an open interval containing c, and if the

limit:

exists, then the line passing through ( c, f(c) ) with slope m is the tangent line to the graph of f at the point ( c, f(c) ).

mtan limx 0

f cx f c x

f(x)

(c, f(c) )

m The blue line is a

Tangent Line with Slope m that also contains the

point ( c, f(c) )

Classifying SlopeDetermine the best way to describe the slope of the

tangent line at each point.

A

B

C

A. Since the curve is decreasing, the slope will also be decreasing. Thus, the slope is Negative.

B. The vertex is where the curve goes from increasing to decreasing. Thus, the slope must be Zero.

C. Since the curve is increasing, the slope will also be increasing. Thus the slope is Positive.

Slope can only be positive, negative, zero, or undefined.

D. Since the curve has a sharp turn, the tangent line will be vertical. Thus the slope is Undefined.

D

29 6 24 8 9 6

0lim x x x

xx

Example 1Find the instantaneous rate of change to at (3,-6).

c is the x-coordinate of the point on the

curve

Direct substitution

mtan limx 0

f 3x f 3 x

limx 0

3x 2 8 3x 9 32 8 3 9 x

limx 0

x 2 2xx

limx 0

x 2

Substitute into the function

0 2

limx 0

x x 2 x

Simplify in order to cancel the

denominator

f x x 2 8x 9

2

limx 0

812x6 x 2 x 3 2x 10x

Example 2Find the equation of the tangent line to at (2,10).

c is the x-coordinate of the point on the

curve

Direct substitution

mtan limx 0

f 2x f 2 x

limx 0

2x 3 2x 23 2 x

limx 0

13x6 x 2 x 3

x

limx 0

136x x 2

Substitute into the function

1360 0 2

limx 0

x 136x x 2 x

Simplify in order to cancel the

denominator

f x x 3 x

13

y 10 13 x 2

Just the slope. Now use the point-slope formula to find the

equation

A Function to Describe SlopeIn the preceding notes, we considered the slope of a

tangent line of a function f at a number c. Now, we change our point of view and let the number c vary by replacing it with x.

mtan limx 0

f cx f c x

The slope of a tangent line at the point x = c.

A constant.

m x limx 0

f xx f x x

A function whose output is the slope of a tangent line at

any x.

A variable.

The slope function or an instantaneous rate of change

function, will be referred to as the Derivative of a Function.

The slope of a tangent line or the instantaneous rate of

change, will be referred to as a Derivative of a function at a

value of x.

The Derivative of a FunctionThe limit used to define the slope of a tangent line is also

used to define one of the two fundamental operations of calculus:

The derivative of f at x is given by

Provided the limit exists. For all x for which this limit exists, f ' is a function of x.

0

( ) ( )'( ) limx

f x x f xf xx

READ: “f prime of x.”

Other Notations for a Derivative:

dydx

'y ( )d f xdx

Vocab

Differentiation: The process of finding the derivative of a function.

Differentiable: The derivative exists. Able to be differentiated. Possess a derivative. Etc.

1 11 10

lim x x xx x x xx

Example 1Derive a formula for the slope of the tangent line to the graph of .

Substitute into the function

Direct substitution

0

' lim f x x f xxx

f x

1 1

11

0lim xx x

xx

1 1

1 10lim x x x

x x x xx

1

1 10lim x x xx

Multiply by a common

denominator

1

1 0 1x x

1 10lim x

x x x xx

Simplify in order to cancel the

denominator

11 xf x

21

1 x

A formula to find the slope of any tangent

line at x.

1 11 1

x x xx x x

Find the Derivative

limx 0

5 xx 1 5 x 1x

Example 2Differentiate .

Substitute into the function

Direct substitution

f ' x limx 0

f xx f x x

limx 0

5 xx 1 5 x 1 x

limx 0

5 xx 5 xx

5 limx 0

xx xx

5 limx 0

xx xx xx x

limx 0

5 xx x x

Simplify in order to cancel the denominator

f x 5 x 1

5 limx 0

xx xx x

xx x

xx x

5 limx 0

1xx x

10

5x x

52 x

Make the problem easier by factoring out common

constants

limx 0

x 3 3x 2 x3x x 2 x 3 x 3

x

Example 3Find the tangent line equation(s) for such that the tangent line has a slope of 12.

Find the derivative first since the

derivative finds the slope for an x value

Find when the derivative equals 12

f ' x limx 0

f xx f x x

3 3

0lim x x x

xx

limx 0

3x 2 x3x x 2 x 3

x

22

0lim 3 3x

x x x x

3x 2 3x0 0 2

limx 0

x 3x 2 3xx x 2 x

f x x 3

3x 2

3x 2 12

x 2 42x

Find the output of the

function for every input

2f 2f

32 32

8

8

8 12 2y x 8 12 2y x Use the point-slope

formula to find the equations

Alternate Definition of a Derivative

The derivative of a function provides us with a measure of the instantaneous rate of change. Thus, we get the derivative at x0 or f'(x0) if we take the limit as the denominator goes to 0:

The slope of a secant line between x and x0 is:

0

0

f x f xx x

0

0

0

limx x

f x f xx x

Known as The Difference Quotient at a particular point

ExampleFind f '(2) if :

f x x 3 xSince the derivative needs to be evaluated at a point, the alternate

definition can be used.

2

2' 2 lim

2x

f x ff

x

3 3

2

2 2lim

2x

x x

x

3

2

10lim2x

x xx

Substitute into the function

Simplify in order to cancel the denominator

2

2

2 2 5lim

2x

x x x

x

Factor

2

2lim 2 5x

x x

22 2 2 5 Direct substitution

13

AP Type Question

Evaluate the limit:

2

2

2

sin sinlimx

xx

This is a difficult limit to evaluate.

0

0

0

limx x

f x f xx x

Notice that it is just applying the

definition of a derivative at a

point

The limit is equal to the derivative of sine at Pi/2. We have not calculated this derivative YET, so we

are currently not able to answer this question.

How Do the Function and Derivative Function compare?

f x 12x

f ' x 112x

Domain: Domain:

12 ,

12 ,

f is not differentiable at x = -½

Differentiability Justification 1In order to prove that a function is differentiable at x = c, you must show the following:

In other words, the derivative from the left side MUST EQUAL the derivative from the right side.

Common Example of a way for a derivative to fail:

limx 0

f cx f c x lim

x 0

f cx f c x

Other common examples: Corners, Cusps

Not differentiable at x = -4

limx c

f x f c

Differentiability Justification 2In order to prove that a function is differentiable at x = c, you must show the following:

In other words, the function must be continuous.

Common Example of a way for a derivative to fail: Other common examples:

Gaps, Jumps, AsymptotesNot differentiable

at x = 0

Differentiability Justification 3In order to prove that a function is differentiable at x = c, you must show the following:

In other words, the tangent line can not be a vertical line.

An Example of a Verticaltangent where the derivative to Fails to exist:

limx 0

f cx f c x

Not differentiable at x = 0

Differentiability Implies Continuity

The contrapositive of this statement is true: If f is NOT continuous at x = c, then f is NOT differentiable at x = c.

The converse of this statement is NOT always true: If f is continuous at x = c, then f is differentiable at x = c.

The inverse of this statement is NOT always true: If f is

NOT differentiable at x = c, then f is NOT continuous at x = c.

If f is differentiable at x = c, then f is continuous at x = c.

Example 1Determine whether the following derivatives exist for the

graph of the function.

1. ' 8

2. ' 6

3. ' 2

4. ' 2

5. ' 3

6. ' 4

f

f

f

f

f

f

DNE

DNE

DNE

Exists

DNE

Exists

Example 2Show that does exist if ' 1f

2

1limx

x

The one-sided derivatives are equal and non-infinite.

2 if 1

2 1 if 1x x

f xx x

1

limx

f x

1

lim 2 1x

x

21Prove that it is Continuous

2 1 1 1

limx

f x

2 21 2 1

0lim x x

xx

2

0lim x x

xx

Since the function is continuous at x=1.

2 21 1

0lim x

xx

The

limit

exis

ts

f(1) e

xist

s1

1 1f 2 1 1 1

1

lim 1x

f x f

1 1 1

lim =1 since lim limx x x

f x f x f x

Prove the Right Hand Derivative is the same as the Left Hand Derivative (and non-infinite)

2 0

lim 2x

x

2 2 1 1

0lim x

xx

2

0lim x

xx

2 1 1 2 1 1

0lim x

xx

20

lim 2x

Thus the derivative exists, at x=1.

Example 3Show that does not exist if .

f x x 3 1

f ' 3

3 2 1

0lim x

xx

First rewrite the absolute value function as a piecewise

function

Since the one-sided limits are not equal, the derivative does

not exist

2 if 34 if 3

x xf x

x x

limx 0

f 3x f 3 x

limx 0

xx

limx 0

1

Find the Left Hand Derivative

1

3 2 3 2

0lim x

xx

3 4 1

0lim x

xx

limx 0

f 3x f 3 x

0lim x

xx

0lim 1x

Find the Right Hand

Derivative

1

3 4 3 4

0lim x

xx

Function v DerivativeCompare and contrast the function and its derivative.

FUNCTION DERIVATIVE-5

-5Vertex

x-intercept

Decreasing

Negative Slopes

Increasing

Positive Slopes

Function v DerivativeCompare and contrast the function and its derivative.

FUNCTION DERIVATIVE

-5 -5

Local Max

x-intercept

Decreasing

Negative Derivatives

Increasing

Positive Derivatives

5 5

Local Min

x-intercept

Positive Derivatives

Increasing

Example 1Accurately graph the derivative of the function graphed

below at left.The Derivative does not exist at a corner.

Make sure the x-value does not have a derivative

The slope from -∞ to

-7 is -2

The slope from -7 to 2

is 0

The slope from 2 to ∞

is 1

Example 2Sketch a graph of the derivative of the function graphed

below at left.

1. Find the x values where the slope of the tangent line is zero (max, mins, twists)2. Determine whether the function is increasing or decreasing on each interval

Increasing

Positive

Decreasing

Negative

Increasing

Positive

Decreasing

Negative

Increasing

Positive

Example 3Sketch a graph of the derivative of the function graphed

below at left.

1. Find the x values where the slope of the tangent line is zero (max, mins, twists)2. Determine whether the function is increasing or decreasing on each interval

Decreasing

Negative

Increasing

Positive

Increasing

Positive