Power RulePower Rule

If H(x) = (3xIf H(x) = (3x77-4x)12/12

then H’(x) = (3x3x7 7 - 4x)11(21x6 - 4)

When differentiating, the first answer is multiplied by the derivative, g’(x).

Power RulePower Rule

If H(x) = (3xIf H(x) = (3x77-4x)12/12

then H’(x) = (3x3x77-4x)11(21x6-4)

H is an antiderivative of (3x3x77-4x)11(21x6-4)

12711 67 (3 4 )

12(3x -4x) (21x -4) x x

Cdx

Anti Power RuleAnti Power Rule

Thus that term must be removed when antidifferentiating.

1n ( ( ) )

1( g(x) ) ( g '(x) )

ng xC

ndx

Anti Power RuleAnti Power Rule

copy

derivative is here

( ) ( )

( )( ) ( ) Cdx

1( ( ) )1

( g(x) ) ( g '(x) )n

n g xC

ndx

Add one

Copy

Anti Power RuleAnti Power Rule

Copy g(x)Copy g(x)

If derivative is hereIf derivative is here

Add one to exponentAdd one to exponent

Divide by new Divide by new exponentexponent

1n ( ( ) )

1( g(x) ) ( g'(x) )

ng xC

ndx

The derivative of (2x-3).The derivative of (2x-3).

4 2 (2x-3) dxWhat is 2?

n( g(x) ) ( g '(x) )dx

The derivative of (2x-3).The derivative of (2x-3).

4 2 (2x-3) dxn( g(x) ) ( g '(x) )dx

n+1

n+1

(g(x) )C

5

5

(2x-3)C

Anti Power RuleAnti Power Rule

Copy g(x)Copy g(x)

If derivative is hereIf derivative is here

Add one to exponentAdd one to exponent

Divide by new Divide by new exponentexponent

1n ( ( ) )

1( g(x) ) ( g'(x) )

ng xC

ndx

==

A.A. (7x+2)(7x+2)1313/13 /13

B.B. (7x+2)(7x+2)1313/13 + C/13 + C

C.C. (7x+2)(7x+2)1111/13 + C/13 + C

D.D. (7x+2)(7x+2)1111/11 + C/11 + C

12(7 2) 7x dx

==

A.A. (7x+2)(7x+2)1313/13 /13

B.B. (7x+2)(7x+2)1313/13 + C/13 + C

C.C. (7x+2)(7x+2)1111/13 + C/13 + C

D.D. (7x+2)(7x+2)1111/11 + C/11 + C

12(7 2) 7x dx

What is the derivative ofWhat is the derivative of12x + 3 ?12x + 3 ?

2

2

12 3dx

x 212 3 2x dx

11 [12 3]

6 1

xC

2112 3 12

6x dx

Number of cable Number of cable telephones 3.2 million in telephones 3.2 million in 20042004 (t=0)(t=0) Rate of growth r(t) = 3.36(t+1)Rate of growth r(t) = 3.36(t+1)0.050.05

How fast is it growing in 2008?How fast is it growing in 2008? How many cable phones in 2008?How many cable phones in 2008?

Rate r(t)=3.36 (t+1) Rate r(t)=3.36 (t+1) 0.050.05. How . How fast is it growing in 2008?fast is it growing in 2008?

r(4) = 3.36(4+1) r(4) = 3.36(4+1) 0.050.05 = = million/yrmillion/yr

r(t)=3.36 (t+1) r(t)=3.36 (t+1) 0.050.05

Find r(4) = Find r(4) = mill./yr.mill./yr.

3.643.64

0.20.2

Number of cable Number of cable telephones 3.2 million in telephones 3.2 million in 20042004 (t=0)(t=0) Rate of growth r(t) = 3.36(t+1)Rate of growth r(t) = 3.36(t+1)0.050.05

How fast is it growing in 2008?How fast is it growing in 2008? How many cable phones in 2008?How many cable phones in 2008?

N(t) = N(t) =

A.A. N(t)= N(t)= ..

B.B. N(t) N(t) ..

C.C. N(t)N(t)

0.053.36 (t+1) dt

1.05( 1)3.36

1.05

tC

1.05( 1)

1.05

tC

-0.953.36 (0.05)(t+1) C

N(t) = N(t) =

A.A. N(t)= N(t)= ..

B.B. N(t) N(t) ..

C.C. N(t)N(t)

0.053.36 (t+1) dt

1.05( 1)3.36

1.05

tC

1.05( 1)

1.05

tC

-0.953.36 (0.05)(t+1) C

Rate r(t)=3.36 (t+1) Rate r(t)=3.36 (t+1) 0.050.05. How . How many phones are there in many phones are there in 2008?2008?

N(t) =N(t) = = 3.36(t+1)= 3.36(t+1)1.051.05/1.05 + C /1.05 + C = 3.2(t+1)= 3.2(t+1)1.051.05 + C + C N(0) = 3.2 + C = 3.2 soN(0) = 3.2 + C = 3.2 so

0.053.36 (t+1) dt

N(0) = 3.2 + C = 3.2N(0) = 3.2 + C = 3.2C = C = millionmillion

N(0) = 3.2 + C = 3.2N(0) = 3.2 + C = 3.2C = C = millionmillion

0.00.0

0.10.1

N(t) = 3.2(t+1)N(t) = 3.2(t+1)1.051.05. How many . How many phones are there in 2008?phones are there in 2008?

N(t) = 3.2(t+1)N(t) = 3.2(t+1)1.051.05

N(4) = 3.2(5)N(4) = 3.2(5)1.051.05

N(4) = million subscribersN(4) = million subscribers

N(t) = 3.2(t+1)N(t) = 3.2(t+1)1.051.05

N(4) = N(4) = millionsmillions

N(t) = 3.2(t+1)N(t) = 3.2(t+1)1.051.05

N(4) = N(4) = millionsmillions

17.3417.34

1.01.0

==2

3

0

[2 3]x dx

==

-10.0-10.0

0.10.1

23

0

[2 3]x dx

What is the derivative ofWhat is the derivative of4x4x22 + 12x + 3 ? + 12x + 3 ?

1

920

2 3

4 12 3

xdx

x x

What is the exponent?What is the exponent?

1 1

9 92 20 0

2 3 1 8 12

44 12 3 4 12 3

x x

x x x x

Now it is in the correct Now it is in the correct form.form.

1 1

2 992

0 0

1 8 12 1(4 12 3) (8 12)

4 44 12 3

xdx x x x dx

x x

Anti Power RuleAnti Power Rule

Copy 4xCopy 4x22+12x+3+12x+3

Derivative is hereDerivative is here

Add one to exponentAdd one to exponent

Divide by new exponentDivide by new exponent

11 2 82 9

00

1 1(4 12 3) |(4 12 3) (8 12)

4 4 ( 8) |

x xx x x dx

12 8

0

1(4 12 3) |

4 ( 8) |

x x

8 8

8 8

19 3 1 10.000004763

32 32 19 32 32(3)

Anti Power RuleAnti Power Rule

Copy 5xCopy 5x33 - 1 - 1

Derivative is hereDerivative is here

Add one to exponentAdd one to exponent

Divide by new exponentDivide by new exponent

2 2

3 1/ 939

2 2 15

15 (5 1)5 1

x xdx dx

xx

3 8/ 93 1/ 9 22 2 (5 1)

(5 1) (15 )15 15 8 / 9

xx x dx C

Anti Power RuleAnti Power Rule

If H(x) = (3xIf H(x) = (3x77-4x)12

H’(x) = 12(3x3x77-4x)11(21x6-4)

12711 67 (3 4 )

12(3x -4x) (21x -4) x x

Cdx

Chain RuleChain Rule

If H(x) =F(g(x)) then H’(x) = F’(g(x)) If H(x) =F(g(x)) then H’(x) = F’(g(x)) g’(x)g’(x)

H(x) is an antiderivative of F ’(g(x)) H(x) is an antiderivative of F ’(g(x)) g’(x)g’(x)

Anti – Chain RuleAnti – Chain Rule

F' g x g ' x dx=F(g(x))+C

Make sure g’(x) is thereMake sure g’(x) is there Antidifferentiate F’ giving FAntidifferentiate F’ giving F Copy g(x) giving F(g(x))Copy g(x) giving F(g(x))

Anti Chain RuleAnti Chain Rule

2 2cos( )[2 ] sin( )x x dx x C

Anti Chain RuleAnti Chain Rule

Antidifferentiate Antidifferentiate cos(x)cos(x)

Derivative of xDerivative of x22 is here is here

Copy xCopy x22

2 2cos( )[2 ] sin( )x x dx x C

Anti Chain RuleAnti Chain Rule

Antidifferentiate Antidifferentiate

sec(x)tan(x)sec(x)tan(x)

Derivative of 9x is Derivative of 9x is herehere

Copy 9xCopy 9x

sec(9 ) tan(9 )[9] sec(9 )x x dx x C

Anti Chain RuleAnti Chain Rule

Derivative of 3x is not Derivative of 3x is not here!here!

cos(3 )x dx

Anti Chain RuleAnti Chain Rule

Now it is in the correct form.Now it is in the correct form.

1cos(3 ) cos(3 )[3]

3x dx x dx

Anti Chain RuleAnti Chain Rule

Antidifferentiate Antidifferentiate

cos(x)cos(x)

Derivative of 3x is here!Derivative of 3x is here!

Copy 3xCopy 3x

1 1cos(3 )[3] sin(3 )

3 3x dx x C

Anti Chain RuleAnti Chain Rule

Antidifferentiate Antidifferentiate

[sec(x)][sec(x)]22=sec=sec22(x)(x)

Derivative of sin(x) is Derivative of sin(x) is herehere

Copy sin(x)Copy sin(x)

2(cos( ))[sec(sin(x))] tan(sin( ))x Cdx x

==

==

/ 6

0

cos(3 )x dx

/ 6

0

1cos(3 )3

3x dx

/ 6

0

1 |sin(3 )

3 |x

1[sin( ) sin(0)]

3 2

1

3

==/ 4

0

cos(2 )x dx

==

0.50.5

0.10.1

/ 4

0

cos(2 )x dx