BAÛNG COÂNG THÖÙC ÑAÏO HAØM - NGUYEÂN HAØM I. Caùc coâng thöùc tính ñaïo haøm. 1. (u

v)'

2. (u.v)'

u' v' 1. ku '

Heä Quaû:

u '.v

u.v ' '

1 2. v

k.u '

u 3. v

'

u '.v

u.v ' v2

v' v2

II. Ñaïo haøm vaø nguyeân haøm caùc haøm soá sô caáp. Bảng đạo hàm x '

x 

 

u '   .u '.u 1

1

sin x  '  cos x

2

x

  x dx 

sin u  '  u '.cos u

 cos x  '   sin x  tan x  '  cos1

Bảng nguyên hàm

 cos u  '  u '.sin u

 1  tan 2 x

 tan u  '  cosu ' u  u '. 1  tan u  2

2

 cot x  '  sin1 x   1  cot x   cot u  '  sinu 'u  u '. 1  cot u  2

2

2

2

u' u.ln a u' ln u ' u

1 x ln a 1 ln x ' x

loga x '

ax '

loga u '

au '

a x . ln a

ex '

u

c

1

 sin xdx   cos x  c

 sin  ax  b  dx   a cos  ax  b   c

 cos xdx  sin x  c

 cos  ax  b  dx  a sin  ax  b   c

1

1

1

1

1

1

dx  tan x  c

 cos  ax  b  dx  a tan  ax  b   c

1  sin2 x dx   cot x  c

 sin  ax  b  dx   a cot  ax  b   c

 cos

2

x

2

2

1

x  a dx 

1

x

x

1

 ax  b dx  a ln ax  b  c a x   c  .ln a 1 ax  b ax  b  e dx  a e  c

ax c ln a

 e dx  e

u

 1



 x dx  ln x  c

a u .u '.ln a

 e  '  u '.e

ex

1  ax  b    ax  b  dx  a .   1

x 1  c,   1  1

 x  a dx 

c

Boå sung: dx x

2

a

2

1 x arctan a a

C

x

III. Vi phaân: dy VD: d(ax d(ln x )

b)

1 x ln 2a x

dx 2

adx

dx , d(tan x ) x

a

2

a a

C

dx a2

x2

arcsin

x a

C

dx x

2

a

2

ln x

y ' .dx dx

1 d (ax a

dx , d(cot x ) cos2 x

b ) , d(sin x ) dx ... sin2 x

cos xdx , d(cos x )

sin xdx ,

x2

a2

C

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

1  ;a a



a 

a .a

a



a

 .

a.b

a

 



a

a

a a .b ; b





a

a

1 : a

0

a







a

a b

1 : a

 a

loga 1

0 ; loga a

a

1

 



x



loga b.c

loga b

b c

loga b

logb c

c

x, 0

logb a

;a



loga c loga 



loga c.logc b

loga b

1 logb a

loga 

loga 

1 : loga 

0

a

 loga b

loga c

loga b

a

1

a

1 ; loga b

loga b ; loga a 

loga b

a



0





M

1

0

aM

loga x

loga





a



a ;  a





a

Haøm soá Logarit

logc b logc a





loga 

1 : loga 





loga 





II.Moät soá giôùi haïn thöôøng gaëp.

1. lim 1 x

1 x

x

2. lim 1  x   e 1 x

x 

a 1 3. lim  ln a x 1  x  4. lim a x x

e

x 0

a

x 0

5. lim x 0

log 1  x   log e x a

a