• 2-1 The Derivative and the Tangent Line Problem• 2-2 Basic Differentiation Rules and Rates of Change• 2-3 Product/Quotient Rule and Higher-Order Derivatives• 2-4 Chain Rule• 2-5 Implicit Differentiation• 2-6 Related Rates (SKIP for now)

Chapter 2 Differentiation

• Basic Differentiation Rules• Product Rule• Quotient Rule• Derivative of Trigonometric

Functions• Higher Order Derivatives

2.3 Product/Quotient Rules and Higher Order Derivatives

Day

1

http://www.youtube.com/watch?v=-chXvU4pza4(Chain, Product, Quotient Rule song :)

[ ( ) ( )] ( ) '( ) ( ) '( )d

f x g x f x g x g x f xdx

In function notation:

Notice that this is not just the product of two derivatives.

1st x the derivative of the 2nd + 2nd x the derivative of the first. (or vice versa)

In words:

Example: Find 2 33 2 5d

x x xdx

2 3x 26 5x 32 5x x 2x

3

2Find if 2 (3 (2 1)).

dyy x x

dx x

You Try…

3 2Rewrite 2 2 (6 3 ).y x x x

3 2 42 2 (12 3) (6 3 )( 6 )dy

x x x x xdx

first function

derivative of second function

derivative of first function

second function

FOIL to check answer with Warm Up #3.

2

( ) ( ) '( ) ( ) '( )[ ]

( ) ( )

d f x g x f x f x g x

dx g x g x

In function notation:

Notice that this is not just the quotient of two derivatives.

Bottom x the derivative of the top – top x the derivative of the bottom all over bottom squared.

In words:

d hi

dx lo

( ) ( ) ( ) ( )

( )

lo d hi hi d lo

lo

2

2bottom

bottomthe

ofderivativetop

topthe

ofderivativebottom

quotienta

ofDerivative

Example: 3

2

2 5Find

3

d x x

dx x

derivative of top function

derivative of bottom function

top function

2 3x 26 5x 32 5x x 2x

22 3x

bottom function squared

Redo Warm Up #1 using Quotient Rule.

bottom function

You Try…

2

3 5Find '( ) if ( ) .

2

xf x f x

x

2

22

3 2 2 3 5( )

2

x x xf x

x

2

22

3 10 6

2

x x

x

http://www.youtube.com/watch?v=-chXvU4pza4(Chain, Product, Quotient Rule song :)

Not every quotient needs to be differentiated by the Quotient Rule.

2

4

2

2

Find '( ) if :

3a) ( )

6

5b) ( )

8

3(3 2 )c) ( )

79

d) ( )5

f x

x xf x

xf x

x xf x

x

f xx

21( 3 )

6x x

45

8x

1'( ) (2 3)

6f x x

35'( ) (4 )

8f x x

3(3 2 )

7x

3'( ) ( 2)

7f x

29

5x

39'( ) ( 2 )

5f x x

You Try…

Differentiate:

1. 4.

2. 5.

3. 6.x

xh3

2)(

)43)(12()( 2 xxxg

13 xxy

x

xxxf

13)(

2

12

x

xxf

2 2(3 1)y x

12

x

xxf

22

2

22

22

22

2

22

22

1

1

1

21

1

211

1

11'

x

x

x

xx

x

xxx

x

xdxd

xxdxd

xxf

Solutions #1

2. Same derivative by expanding and using the Power Rule.

31618)(

4386)(

)43)(12()(

2

23

2

xxxf

xxxxf

xxxf

1. The Product Rule

31618)(

161236)(

4)43(3)12()(

12)43(43)12()(

2

22

2

22

xxxf

xxxxf

xxxxf

xdx

dxx

dx

dxxf

)4x3)(1x2()x(f 2

Solutions #2

Product Rule:

Find f’(x) for

22

522

5

2

5

22

1

2

13

2

13

32

733

2

1)(

3102

1)(

Rule.ProducttheuseNow

1)(

form.lexponentiainradicaltherewriteFirst

xxxxxxf

xxxxxf

xxxf

13 xxxf

Solutions #4

Quotient Rule:

2

2

2

22

2

2

11332

1)13(32)('

x

x

x

xxxx

x

xxxxxf

Solutions #5

x

xxxf

13)(

2

2

2

2221

122

11111013

.derivativethefindNow

31313

x

x

xxxxx

dx

d

xxxx

x

x

x

x

xx

division first and then power rule:

Using Derivative Rules

Suppose u and v are functions that are differentiable atx = 3, and that u(3) = 5, u’(3) = -7, v(3) = 1, and v’(3)= 4.Find the following at x = 3 :

)(.1 uvdx

d'')( vuuvuv

dx

d 8)7)(1()3(5

v

u

dx

d.2

2

''

v

uvvu

v

u

dx

d

21

)4)(5()7)(1( 27

u

v

dx

d.3

2

''

u

vuuv

u

v

dx

d

25

)7)(1()4)(5( 25

27

Closure

Explain how to use the quotient and product rules to find a derivative.

• Basic Differentiation Rules• Product Rule• Quotient Rule• Derivative of Trigonometric

Functions• Higher Order Derivatives

2.3 Product/Quotient Rules and Higher Order DerivativesD

ay 2

2

0

2

Consider the function siny

We could make a graph of the slope: slope

1

0

1

0

1Now we connect the dots!The resulting curve is a cosine curve.

sin cosd

x xdx

Derivatives of Trigonometric Functions

Proof

h

xhxx

dx

dh

sin)sin(limsin

0

h

xxhhxx

dx

dh

sincossincossinlimsin

0

h

xh

h

hxx

dx

dhh

cossinlim

)1(cossinlimsin

00

h

xhhxx

dx

dh

cossin)1(cossinlimsin

0

= 0 = 1

xxdx

dcossin

Slope of y = cos x

The curve y´ = –sin x as the graph of the slopes of the tangents to the curve y = cos x.

h

xhxx

dx

dh

cos)cos(limcos

0

h

xxhhxx

dx

dh

cossinsincoscoslimcos

0

h

xh

h

hxx

dx

dhh

sinsinlim

)1(coscoslimcos

00

h

xhhxx

dx

dh

sinsin)1(coscoslimcos

0

Proof

= 0 = 1

xxdx

dsincos

Find the derivative of f(x) tan x.

tand

xdx

sin

cos

d x

dx x

2

cos cos sin sin

cos

x x x x

x

2 2

2

cos sin

cos

x x

x

2

1

cos x

2sec x

2tan secd

x xdx

Challenge:

sin cosd

x xdx

cos sind

x xdx

2tan secd

x xdx

2cot cscd

x xdx

sec sec tand

x x xdx

csc csc cotd

x x xdx

Find the derivatives of the 3 remaining trig functions.

Trig Review:

Pythagorean Identities

Double Angle Formulas

1cossin 22 xx

xx 22 sectan1

xx 22 csccot1

cossin22sin

22 sincos2cos

Trig Review Double Angle Formulas

22 sincos2cos

2

22

sin212cos

sin)sin1(2cos

1cos22cos

)cos1(cos2cos2

22

Example 2: Find if cos .dy

y x xdx

( sin ) cosdy

x x xdx

Find the derivative of

21( ) 5sin sec tan 7 3

2f x x x x x x

21( ) 5cos sec tan sec tan (1) 14

2f x x x x x x x x

Example 1:

1 sin( )

cos

xf x

x x

2

( cos ) (1 sin ) (1 sin ) ( cos )( )

( cos )

d dx x x x x x

dx dxf xx x

2

( cos )(cos ) (1 sin )(1 sin )( )

( cos )

x x x x xf x

x x

2 2 2 2

2 2

( cos cos ) (1 sin ) cos cos 1 sin( )

( cos ) ( cos )

x x x x x x x xf x

x x x x

2

cos( )

( cos )

x xf x

x x

Ex 3:Differentiate:

Given

For what values of x does the graph of f have

a horizontal tangent?

sec( )

1 tan

xf x

x

Example 4

2

2

2

2 2

2

2

(1 tan ) (sec ) sec (1 tan )'( )

(1 tan )

(1 tan )sec tan sec sec

(1 tan )

sec (tan tan sec )

(1 tan )

sec (tan 1)

(1 tan )

d dx x x x

dx dxf xx

x x x x x

x

x x x x

x

x x

x

Example 4 Solution

Since sec x is never 0, we see that f ’(x)

when tan x = 1. This occurs when x = nπ + π/4, where n is an

integer.

Example 4 Solution

= 0

sec( )

1 tan

xf x

x

x = nπ + π/4

1. Calculate

2. Find the tangent and normal lines to y = x2 sin x at x = 3.

3. Find

You Try…

.cossin xxdx

d

x

xx

dx

d

cos

sin2

1.

Find the tangent and normal lines to y = x2 sin x at x = 3.

dy

dx 2x cos x sin x 2x

2 cos 2 sin x x x x

(3) y 23 sin 3 1.270

(3) y 23 cos 3 2 3 sin 3 8.063 (slope)m

= 8.063 3 +1.270Tangent : y x

= 8.063 + 25.460y x

2.

1= 3 +1.270

8.063Normal :

y x

= 0.124 + 0.898y x

Higher DerivativesThe second derivative of a function f is the derivative of the derivative of f at a point x in the domain of the first derivative.

Derivative Notations

nf

f

f

(4)f

Second

Third

Fourth

nth

2

2

d y

dx3

3

d y

dx4

4

d y

dxn

n

d y

dx

Example 5: 5 3( ) 3 2 14f x x x Given find ( ).f x

4 2( ) 15 6f x x x

3( ) 60 12f x x x

2( ) 180 12f x x

Example 6

Find if . )4/(// f xxf sec)(

xxxf tansec)(/

xxxxxxf tansectansecsec)( 2//

xxxxf sectansec)( 23//

)4/sec()4/(tan)4/(sec)4/( 23// f

23212)4/( 23// f

You Try…

.cos if Find 1. yy

2. Given

3. Compute .

4.

2 1( )

3 2

xf x

x

find (2).f

Find the 27th derivative of cos x.Challenge:

4

4

1d

dx x

cot.

1 cotFind for

dy x

ydx x

1..cos if Find yy

y cossin

y cos

sin cos 1

sin 1 sin sin2cos

Given2 1

( )3 2

xf x

x

find (2).f

3 3

42 42 21(2)

3243(2) 2f

2.

22

2

)4129(

)1218)(7()0)(4129()("

xx

xxxxf

2

2 2

2 3 2 3 2 1 7( ) 7 3 2

3 2 3 2

x xf x x

x x

4129

72

xx

22 ))43((

))43(6(7

x

x3)43(

42

x

4

4

1d

dx x

44 5

4 5

1 246 24

d dx x

dx x dx x

21dx

dx x

22 3

2

12

d dx x

dx x dx

33 4

3

12 6

d dx x

dx x dx

3.

Ex 4:cot

.1 cot

Find for

dy xy

dx x

dy

dx 1 cot x 2csc x cot x 2csc x

21 cot x

dy

dx 2 2csc csc cot x x x 2csc cot x x

21 cot x

2csc x

21 cot x

=

Find the 27th derivative of cos x. The first few derivatives of f(x) = cos x

are as follows:

Therefore, f (24)(x) = cos x f (27)(x) = sin x

(4)

(5)

'( ) sin

''( ) cos

'''( ) sin

( ) cos

( ) sin

f x x

f x x

f x x

f x x

f x x

Challenge:

Closure

Give a way of remembering the derivativesfor the six trigonometric functions.