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