Examples A: Derivatives Involving Algebraic Functions.

Examples A: Derivatives Involving Algebraic Functions

The derivative of composite functionfor the case f(x) = gn(x)

Let:f(x) = gn(x)Then:f' (x) = ngn-1(x) . g'(x)

Example:Let f(x) = (3x8 - 5x + 3 )20

Then f(x) = 20 (3x8 - 5x + 3 )19 (24x7 - 5)

Examples (1)

36

7

5 9

9

7

5 9

9

5 99 59

5 99 599

9

45

32)(.3

32)(.2

1111)(.1

:

xx

xxxf

xxxf

xxxxxx

xxxf

functionsfollowingtheofeachofderivativetheFind

379 3)1()(.7 xxxxxf

73

3)(.6

12

2)(.5

12)(.412)(.4

9

24

4 58

4 584 58

xx

xxxf

xxxf

xxxfxxxf

514

914

54

94

910

98

59

95

59

95

91

91

5

9

9

5

5

99

5

9

1

9

199)(

11

11)(.1

108

99

5 99 5

5 9

9 5

9

99

9

xxx

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xxxxxxxx

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xxf

514

59

59

5

2718327)(

32

32)(.2

86

9

79

7

5 9

9

xxxxxf

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514

59

59

59

5

271832745

43045332)(

4532

45

32)(.3

86

9316

252167

9

3167

9

36

7

5 9

9

xxxxxx

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116124

5)(

12

12)(.4

78

8

4 58

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xxxxf

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5)(

122

12

2)(.5

78

8

4 58

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45

xxxxf

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29

82439

9

24

73

)39)(3()212)(73()(

73

3)(.6

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]3

139)1()1(7[

]3)1([2

1)(

]3)1([

3)1()(.7

3

28769

2

1

3

179

2

1

3

179

379

xxxxx

xxxxxf

xxxx

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Example (2)

28)7(4)2())1(

)2()1()2(

)2())2(()2(

)())(()(

))(()(

:

)2(

7)2(,1)2(&4)1(

))(()(:

hg

hgf

hhgf

xhxhgxf

xhgxf

Soluion

fFind

hhg

xhgxfLet

Examples B: Derivatives Involving Trigonometric Functions

Basic Formulas

xxyxy

xxyxy

xyxy

xyxy

xyxy

xyxy

cotcsccsc.5

tansecsec.5

csccot.4

sectan.3

sincos.2

cossin.1

2

2

General Formulas (Chain Rule)Let u=u(x)

uuuyuy

uuuyuy

uuyuy

uuyuy

uuyuy

uuyuy

cotcsccsc.5

tansecsec.5

csccot.4

sectan.3

sincos.2

cossin.1

2

2

Examples (1)

)!()1(csccos

sin

coscot;coscsccos

sincotcsccos

)sin(cot)csc(cos

cotcos.3

)sec(

)sectansec1(tansec)sec(

sec

tan.2

2sin82cos2

82sin22cos

2sin.1

2

2

2

2

2

2

78

78

8

timprovemenmuchnotxx

x

xxbecausexxx

xxxx

xxxxy

xxy

xxx

xxxxxxxxxy

xxx

xy

xxxx

xxxxy

xxy

Examples (2)

192020

20

19202

20

1920

20

20tansec

sec.3

20sec

tan.2

20sin

cos.1

xxxy

xy

xxy

xy

xxy

xy

Examples (3)

xxy

xxy

xxxy

xxy

xxy

xxy

xxy

xxy

cossin20

)(sinsin.4

tansecsec20

)(secsec.3

sectan20

)(tantan.2

)sin(cos20

)(coscos.1

19

2020

19

2020

219

2020

19

2020

Examples (4)

21

2

22

78

78

8

11:

61

cot

cscsin5cos561

cot

)!(6

1cot

sin5.2

5sin85cos5

85sin55cos

)!(5sin.1

xx

soandxx

Note

xx

xxxxx

xy

quotient

xx

xy

xxxx

xxxxy

productxxy

2

1

2

1

2

1993

1003

2

1:

)2

1cotcsc3()csc3(100

)!()csc3(.3

xxsoandxxNote

xxxxxy

Powerxxy

]tan

sec

3

402[

])(tan432[10

]5sec)(tan3

242[

])(tan432[10

])(tan432[

)tan432(

)tan432(.4

3 5

524

93

25

4523

15

93

25

103

25

1053

2

103 52

x

xx

xxy

xxx

xxy

xx

xx

xxy

Examples (6)

)18()]32cot()32csc([)]32[csc(5

7

)]32[csc()32(csc

)32(csc.2

)18()32(5

7)32cot()32csc(

)32csc(

)32(csc.1

3445

24

5

7445

7

5 47

35

245

745

74

5

74

5 74

xxxxxxxy

xxxx

xxy

xxxxxxxy

xx

xxy

Examples (7)

))18(tan(sin.3

)8(tansin.2

)8sin(tan.1

359

59

5

xy

xy

xy

4

52

5

5

40

8sec

)8cos(tan

)8sin(tan.1

x

x

xy

xy

4

52

5

85

95

59

40

8sec

)8cos(tan

)]8[sin(tan9

)]8[sin(tan

)8(tansin.2

x

x

x

xy

x

xy

4

65

752

75

875

975

759

40

)18(7

)18(sec

])18tan(cos[

])18tan(sin[9

])18tan(sin[

])18tan([sin.3

x

x

x

x

xy

x

xy