課件分析成果review about rules of derivatives

Sum,differenceandconstantrulesofderivatives

f

f

;

;

d

cf

;

d

導(dǎo)數(shù)的加減數(shù)乘運算

Example1:Calculatethederivativeoff

Solution:

初等函數(shù)的導(dǎo)數(shù)

導(dǎo)數(shù)的加減數(shù)乘運算

Example2:Calculatethederivativeofg

Solution:

初等函數(shù)的導(dǎo)數(shù)

Productruleofderivatives

d

Example1:Findthederivativeofy

Solution:Since y

y

導(dǎo)數(shù)的乘法運算

,then

導(dǎo)數(shù)的加減數(shù)乘運算

初等函數(shù)的導(dǎo)數(shù)

3

Example2:Findthederivativeofy2x1x21.

3

導(dǎo)數(shù)的乘法運算

Solution:Since

y2x1x21,then

3 '

3

3 '

y'2x1x212x1'x212x1x21

3

3

2'x1'x212x1x21'

31

3'

導(dǎo)數(shù)的加減數(shù)乘運算

初等函數(shù)的導(dǎo)數(shù)

1 11

x

2x212x1 x23x2

x2x2

2 2

QuotientRuleofderivatives

fx' f'xgxfxg'x

gx

g2x

Example1:computethederivativeofSolution:

fx

x

1x2.

numerator'denominatornumeratordator

'

x '

x'1x2x1x2'

導(dǎo)數(shù)的除法運算

fx

1x2

1x22

denominator2

導(dǎo)數(shù)的加減數(shù)乘運算

x'1x2x1'x2'

初等函數(shù)的導(dǎo)數(shù)

1x22

11x2x2x

1x22

1x2

1x22

Example2:Findthederivativeof

fx

3x2x2

4x31 .

Solution:Applythequotientruleofderivatives,

nume den num

dator

導(dǎo)數(shù)的除法運算

en

導(dǎo)數(shù)的加減數(shù)乘運算

初等函數(shù)的導(dǎo)數(shù)

ThechainRuleofderivatives

d

E

Ex:exterior外部的

In:interior內(nèi)部的

Example1:If

fx

,thenfind

f'x.

確定內(nèi)外層函數(shù)

外層導(dǎo)數(shù)乘以內(nèi)層導(dǎo)數(shù)

Solution:(I) .

(II)

Example2:If

fx

,thenfind

f'x.

確定內(nèi)外層函數(shù)

外層導(dǎo)數(shù)乘以內(nèi)層導(dǎo)數(shù)

Solution:(I)f

(II)

Example3:If

fx

,thenfind

f'x.

確定內(nèi)外層函數(shù)

Solution: (I)f

(II)

cos

xx'cos

x1

Ex

1

In

1

x

外層導(dǎo)數(shù)乘以內(nèi)層導(dǎo)數(shù)

2

Ex

In

Example4:If

fxex,thenfind

f'x

1'

1'

確定內(nèi)外層函數(shù)

Solution:(I)f'xexex

外層導(dǎo)數(shù)乘以內(nèi)層導(dǎo)數(shù)

ExIn

1 1' 1

(II)

exlne exx2

x In

Ex In Ex

Example5:If

fx

16x

,thenfind

f'x

Ex '

確定內(nèi)外層函數(shù)

Solution:(I)f'x

16x'16x

In

外層導(dǎo)數(shù)乘以內(nèi)層導(dǎo)數(shù)

1

Example6:If

Ex

2cos

fx x,thenfind

2

Ex

f'x.

In

x'

x

Solution: (I)

f'x2cos

確定內(nèi)外層函數(shù)

'

2cos

2

Ex

2

外層導(dǎo)數(shù)乘以內(nèi)層導(dǎo)數(shù)

In

xx'

x1 x

(II)

2sin 2sin

sin

Ex 22

In

Ex

22 2

In

Example7:If

fxlnlnx,thenfind

f'x.

確定內(nèi)外層函數(shù)

'

Solution: (I)

f'xlnlnx'lnlnx

外層導(dǎo)數(shù)乘以內(nèi)層導(dǎo)數(shù)

ExIn

1 ' 11

(II)

lnx

lnx In

lnxx

Example8:If

Ex

fxxlnx2,thenfind

Ex In

導(dǎo)數(shù)乘法運算

f'x.

Solution: (I)

f'xxlnx2'x'lnx2xlnx2'

2

'

(II)

lnx

xlnx

ExIn

確定內(nèi)外層函數(shù)

外層導(dǎo)數(shù)乘以內(nèi)層導(dǎo)數(shù)

2

lnx2x1x2'lnx2x12xlnx22

x2 x2

InEx

InEx

Example9:If

fxx

,thenfind

f'x.

1x2

Solution: (I)

f

x

x

1x

'

2'

'

2'

1x2

1x2

導(dǎo)數(shù)乘法運算

確定內(nèi)外層函數(shù)

x

x

1x

Ex '

(II)

x1x2

1x2

外層導(dǎo)數(shù)乘以內(nèi)層導(dǎo)數(shù)

In

(III)

x11x21x

2'

x2

1x2

1x2

21

2 In

Ex

Example10:If

fxx12x3,thenfind

f'x.

Solution: (I)

f'xx12x3'x'12x3x12x3'

導(dǎo)數(shù)乘法運算

確定內(nèi)外層函數(shù)

3'

12x3x12xEx

In

外層導(dǎo)數(shù)乘以內(nèi)層導(dǎo)數(shù)

12x3x312x212x'12x36x12x2

Example11:If

fx

e2x

,thenfind

導(dǎo)數(shù)除法運算

2x

Ex In

f'x.

e2x'

e2x'2xe2x2x'

Solution: (I)

f'x

2x

2x2

確定內(nèi)外層函數(shù)

'

e2x2xe2x2

(II)

Ex

In

外層導(dǎo)數(shù)乘以內(nèi)層導(dǎo)數(shù)

2x2

(III)

e2x2x'2xe2x2

Ex In e

2x2

2x4x24x2

e2x2x1

2x2

Example12:If

fxx12

ex2

2

,thenfind

f'x.

導(dǎo)數(shù)加法運算

3

3 ex2' 3' ex2'

Solution:(I)

f'xx12 x12

確定內(nèi)外層函數(shù)

3'

2

x2'

2

2

eIn

(II)

x1Ex

2

In

外層導(dǎo)數(shù)乘以內(nèi)層導(dǎo)數(shù)

Ex

3 1

'ex2

' 3 1

ex2

(III)

2x22x1

x2 x22

2 2 2

In

Ex

'

InEx

fghx f'gxg'hxh'x

ExMidIn Ex

Mid In

Example13:If

fxlnsinx2,thenfind

f'x.

Solution: (I)

f'xlnsinx2'lnsinx2

確定內(nèi)中外層函數(shù)

'

ExMidIn

(II)

1 cosx22x

外層導(dǎo)數(shù),中層導(dǎo)數(shù),內(nèi)層導(dǎo)數(shù)相乘

sinx2

III

III

Example14:If

ycos23x,thenfind

dy.

dx

確定內(nèi)中外層函數(shù)

Ex

dcos23x

dy dcos23x

Mid In

dx dx

dx

外層導(dǎo)數(shù),中層導(dǎo)數(shù),內(nèi)層導(dǎo)數(shù)相乘

2cos3