Product and Quotient Rule

By: Jenna NeilLori Hissick

And Evan Zimmerman

Period 4

4429 2 xx

Product Rule

• Derivative times uv = u times the derivative of v plus derivative of u

d uv udv vdu( )

Product Rule

• Example: U

VY=

)43)(7( 2 xx

43426 22 xxx

4429 2 xx

)1)(43()6)(7( 21 xxxY

Quotient Rule

• Lo De Hi – Hi De Lo all over Lo squared • U = numerator • V = denominator

vdu udv

v

2d u

v( )

Quotient Rule

• Example:

*Leave the bottom the way it is!*

2 5

3

3

2

x x

x

22

322

)3(

)2)(52()56)(3(

x

xxxxx

22

24

)3(

15132

x

xx

And the game’s begin…

yx

6

3

#1)

#1 Answer) yx x

12

6

6 9

• #2)

yx

x x

4

2

sin

cos

#2 Answer)

Yx x x

x x1

2

8 4

2

( co s sin )

( co s )

• #3)

Yx

x

2

3 1#3 Answer)

)13(

51

x

Y

• #4)

Yx

x

20 4

8

#4 Answer)

Yx

12

164

8

( )

• #5)

Yx

37 620

4

,

#5 Answer)

Yx

12

37 620

4

,

( )

• #6)

Y x x ( )( )9 4 102

#6 Answer)

Y x x1 212 72 10

• #7)

Y x x ( )( )3 6 94 3

#7 Answer)

Y x x1 6 3126 108

• #8)

Y x x x ( )( )3 7 2 124 2

#8 Answer

Y x x x x1 4 3 230 144 42 168

• #9)

#9 Answer

Y (x 3 7x 1)(5x 2)

Y1 20x 3 70x 6x 2 9

• #10)

Y x 2(4 3x 3)

#10 Answer

Y1 15x 6

8

x 3

• #11)

Y x 3 ln x#11 Answer

Y1 x 2(1 3ln x)

• #12)

#12 Answer

Y1 3x1

2 (2x sec2 x 3x tan x)

Y 6x3

2 tan x

• #13)

#13 Answer

Y1 60x 4 59 14(5x 2 2)2

Y 4x 3 7x5x 2 2

• #14)

Y 7x 2

4ex x#14 Answer

Y1 7x(8ex 4xex x)

(4ex x)2

• #15)

Y 2

x 1#15 Answer

Y1 2

(x 1)2

• #16)

#16 Answer

Y1 x(3x 2)(3x 1)2

Y x 2

3x 1

• #17)

Y 4sin x

2x cos x#17 Answer

Y1 8(x cos x sin x) 4(2x cos x)2

• #18)

Y x 2 6x 39x 2 9x 4

#18 Answer

Y1 9x 3 53x 2 50x 3(9x 2 9x 4)2

• #19)

Y 6x 2 2x 53x 2 3x 1

#19 Answer

Y1 12x 2 18x 13(3x 2 3x 1)2

• #20

Y 7x 2 8x3x 2 6x

#20 Answer

Y1 150x 2

(3x 2 6x)2

• #21)

Y 9x 3 4x5x 2 9x

#20 Answer

Y1 45x 4 162x 3 20x 2

(5x 2 9x)2

• #22)

Y 4x 2

12x 4

#22 Answer

Y1 99x 5

12x 6

• #23)

Y 7x 2 6x 43x 3 2x

#23 Answer

Y1 21x 4 36x 3 22x 2 8

(3x 3 2x)2

• #24)

Y 6x 3 5x 2

5x 4

#24 Answer

Y1 30x 6 50x 5

5x 6

• #25)

Y 5x 2 sin x cos x

#25 Answer

Y1 5x 2 cos x sin x 10x sin x cos x

• #26)

Y x(3x 5)sin x

#26 Answer

Y1 3x 2 cos x 5x cos x 6x sin x 5sin x

• #27)

Y x 4 sin x

#27 Answer

Y1 x 4 cos x 4x 3 sin x

• #28)

4

2

12

4

x

xY

#28 Answer

6

51

12

99

x

xY

• #29)

23

2

76

43

xx

xxY

#29 Answer

223

2341

)76(

284818

xx

xxxY

• #30)

)310)(272( 523 xxxxY

#30 Answer

6632410010350100 2345671 xxxxxxY

)310)(272( 523 xxxxY

• #31)

5104512 231 xxxY

)25)(13( 22 xxxY

#31 Answer

• #32)

)4)(( 32 xxY

#32 Answer

xxxY 83 341

19104177224 231 xxxY

• #33)

)358)(237( 22 xxxxY

#33 Answer

• #34)

789814740 231 xxxY)392)(825( 22 xxxxY

#34 Answer

789814740 231 xxxY

• #35)

#35 Answer

49806324 231 xxxY

)833)(352( 22 xxxxY

• #36)

4112812632 231 xxxY)582)(254( 22 xxxxY

#36 Answer

4112812632 231 xxxY

• #37)

)553)(676( 22 xxxxY

#37 Answer

6516615372 231 xxxY

• #38)

)82)(74( 22 xxxxY

#38 Answer

xxxY 11213832 231

• #39)

)7)(6( 43 xxY )7)(6( 43 xxY #39 Answer

61 294xY

• #40)

22

21

)54(

2

xx

xY

xx

xxY

54

762

2

#40 Answer