BAÛNG COÂNG THÖÙC ÑAÏO HAØM - NGUYEÂN HAØM

I. Caùc coâng thöùc tính ñaïo haøm.

'

u v '.

u v . '

v

)'

u

'

u v '.

. '

u 1. (

v '

u v 2.( . )'

u v 3.

2

u v

v

'

ku

'

'

Heä Quaû: 1.

k u 2. .

1 v

'v 2 v

II. Ñaïo haøm vaø nguyeân haøm caùc haøm soá sô caáp.

Bảng nguyên hàm

Bảng đạo hàm

  1

1

ax b 

x

x

 u

'

'

  1 u u . '.

 x dx

 

c ,

 

 1

c

.

 ax b dx

  1 x 

1

1 a

 1

c

ax b dx 

 

ax b 

sin

cos

1 a

xdx

 

x c 

sin

cos

sin

cos

u

u

u

x

sin

'

'.cos

 x '

c

ax b dx 

ax b 

cos

sin

x

xdx

 

x c 

cos

'

sin

cos

sin

x

u

u

u

1 a

 

cos

'

'.sin

2

2

u

u

u

tan

'

x

x

 

tan

'

1 tan

' 2

  '. 1 tan

dx

c

ax b 

tan

2

dx

u cos

x c 

tan

x

1 a

1 ax b 

cos

x

1 2 cos

u

1 2 cos

2

2

dx

c

 

ax b 

cot

x

x

cot

'

u

u

u

 

cot

'

2

    1 cot

  '. 1 cot

1 a

1 ax b 

sin

 1 2 sin

' u

u  2 sin

dx

 

x c 

cot

x

1 2 sin

x

a x

dx

x

c

dx

ax b

c

 

ln

ln

1 a

1 x

1 ax b 

log ' u ' log a x a u a

x

u

x

x

a

'

a

. ln

a

a

'

u a u .

'.ln

x a dx

ln x ' ln u ' 1 ln 1 x u ' .ln u ' u

x    a

dx

c

a

a

x

u

u

x

ax b 

ax b 

e

x e dx

e

c

u e '.

e

'

dx

e

e

c

x e

'

x    a  .ln 1 a

Boå sung:

dx

2

2

dx

dx

dx

ln

x

x

a

C

arctan

C

arcsin

C

ln

C

2

2

2

2

2

2

2

2

x

a

1 a

x a

x a

x

a

x x

a a

1 a 2

x

a

a

x

dy

III. Vi phaân:

y dx ' .

d ax (

b

)

adx

dx

d ax (

b

)

d

cos

sin

VD:

, (sin ) x

xdx , (cos ) x d

xdx ,

1 a

c   a a ln

d

(tan ) x

d

(cot ) x

,

,

. . .

dx 2 sin

dx 2 cos

x

x

d x (ln ) dx x

BAÛNG COÂNG THÖÙC MUÕõ - LOGARIT I. Coâng thöùc haøm soá Muõ vaø Logarit.

Haùm soá muõ

Haøm soá Logarit

M

log

,

a

x

a

0

0

1

  b

a x M x 0

1

1

;

;

b

loga a

loga

log a

log a

 

b

 

a

b ;

loga

 a

loga a

;

a

1  a

b

c

1 log   a b c log . a

log a

log a

 

 

 

a

.a a

a

;

a a

b

c

log a

log a

log a

b c

a

.a 

a

c

a

log b

log b

logaa

 

a

c

;

log c

a b .

  a b ; .

b

b

log a

log .log c c

a

a b

a b 

b a

log c

b

log a

a

a

a

  

0

1

a   

log a

1 log b log a

a

1

  

a

:

a

a

  

1

log a

a

0

1

a

:

a

a

  

0

1

  

: log a : log a

log a

II.Moät soá giôùi haïn thöôøng gaëp.

x

1

a x

x

log

a

e

log

lim.5

lim.3

ln

a

a

x

0

x

0

e

1

1

. lim x

 1 x

1

 1

lim.4

a

x

0

lim.2

e

 1

1 x  x x

x



 x  x a  x