Power Rule If H(x) = (3x 7 If H(x) = (3x 7 -4x) 12 /12 3x 7 then H’(x) = (3x 7 - 4x) 11 (21x 6 -...

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


copy

derivative is here

( ) ( )

( )( ) ( ) Cdx

1( ( ) )1

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

n g xC

ndx

Add one

Copy


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


Copy g(x)Copy g(x)

If derivative is hereIf derivative is here


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

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

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


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

Derivative is hereDerivative is here


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)


Copy 5xCopy 5x33 - 1 - 1

Derivative is hereDerivative is here


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


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


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


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


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

cos(3 )x dx


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

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

3x dx x dx



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



[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

Power Rule If H(x) = (3x 7 If H(x) = (3x 7 -4x) 12 /12 3x 7 then H’(x) = (3x 7 - 4x) 11 (21x 6 -...

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Transcript of Power Rule If H(x) = (3x 7 If H(x) = (3x 7 -4x) 12 /12 3x 7 then H’(x) = (3x 7 - 4x) 11 (21x 6 -...