chuyên đề đạo hàm (hot)

Chuyªn ®Ò I: §¹o hµm

A- Tãm t¾t lý thuyÕt

I) Hµm sè liªn tôc

+ Cho hµm sè y = f (x) x¸c ®Þnh trªn kho¶ng (a;b) . Hµm sè y = f (x) ®îc

gäi lµ liªn tôc t¹i x (a b) o ∈ ; ⇔

o

x→x

lim f(x) = ( ) xo

f

+ Chó ý : Hµm sè y = f (x) x¸c ®Þnh trªn kho¶ng (a;b) liªn tôc t¹i x (a b) o ∈ ;

⇔ ∃ → −

o

x x

lim f(x) vµ → +

o

x x

lim f(x)

→ −

o

x x

lim f(x) = → +

o

x x

lim f(x) = ( ) xo

f

II) §¹o hµm

1) §Þnh nghÜa

+ Cho hµm sè y = f (x) x¸c ®Þnh trªn tËp x¸c ®Þnh cña nã vµ xo ∈ TX§

®¹o hµm cña hµm sè y = f (x) t¹i xo kÝ hiÖu ( )

'

o

y x hay ( )

'

o

f x lµ

( )

'

o

y x = ( )

'

o

f x = o

x→x

lim

x

y

∆

∆

=

o

x→x

lim

x

f x x f x o o

∆

( + ∆ ) − ( )

( ) ( )

o o ∆y = y − y = f x − f x gäi lµ sè gia t¬ng øng cña h/s t¹i xo

o ∆x = x − x gäi lµ sè gia cña ®èi sè t¹i xo

+ Hµm sè y = f (x) x¸c ®Þnh trªn tËp x¸c ®Þnh cña nã vµ xo ∈ TX§

⇔ ∃ ( )

−

o

f x

'

vµ ( )

+

o

f x

'

( )

−

o

f x

'

= ( )

+

o

f x

'

= ( )

'

o

f x

Víi ( )

−

o

f x

'

= → −

o

x x

lim

x

y

∆

∆

vµ ( )

+

o

f x

'

= → +

o

x x

lim

x

y

∆

∆

+ Chó ý : H/s y = f (x) cã ®¹o hµm t¹i xo th× nã liªn tôc t¹i xo ngù¬c l¹i

th× cha ch¾c

2) Ph ¬ng tr×nh tiÕp tuyÕn

Cho H/s y = f (x) (C) vµ M( xo

yo

; ) ∈(C)

Ph¬ng tr×nh tiÕp tuyÕn t¹i M lµ: ( ) o o o

y − y = f (x ) x − x

'

3) C¸c quy t¾c tÝnh ®¹o hµm

' ' '

(u ± v) = u ± v

' '

(ku) =k.u víi k lµ h»ng sè

2

' '

'

v

u v v u

v

u −

 =











uv u v v u

' ' '

( ) = +

' 1` '

(u ) .u .u

−

=

α α

α α ∈R

2

'

'

1

u

u

u

 = −











y = f(u) vµ u = g(x) th× ' ' '

.

x u

ux

y = y

4) B¶ng ®¹o hµm

1

§¹o hµm c¸c hµm sè s¬ cÊp c¬ b¶n §¹o hµm c¸c hµm sè hîp

' 1

( ) .

−

=

α α

x α x

' 1 '

(u ) .u .u

−

=

α α

α

2

'

1 1

x x

 = −











( )

x

x

2

' 1

=

2

'

'

1

u

u

u

 = −











u

u

u

2

( )

'

'

=

Sinx =Cosx '

( )

(Cosx) =−Sinx '

( ) tg x

Cos x

Tgx 2

2

'

1

1

= = +

( ) ( Cotg x)

Sin x

Cotgx 2

2

'

1

1

= − = − +

( Sinu) u .Cosu ' '

=

(Cosu) u .Sinu ' '

=−

( ) .(1 )

' 2

2

'

'

u Tg u

Cos u

u

Tgu = = +

( ) (1 )

' 2

2

'

'

u Cotg u

Sin u

u

Cotgu = − = − +

( )

x x

e =e

'

(a ) a a

x x

.ln '

=

( )

x

x

1

ln '

=

( )

x a

x a

ln

1

log '

=

( )

u u

e u .e

'

'

=

(a ) u a a

u u

. .ln '

'

=

( )

u

u

u

'

'

ln =

( )

u a

u

a

u

.ln

log

'

'

=

5) Vi ph©n

Cho H/s : y = f(x) x¸c ®Þnh trªn kho¶ng (a;b) vµ cã ®¹o hµm t¹i x∈(a;b)

Khi ®ã dy y .dx '

=

B- Bµi tËp tr¾c nghiÖm

1) §¹o hµm cña hµm sè 3 2 1

4 2

y = x − x − t¹i x0 =1 b»ng

A 8 B 6 C 3 D 0

2) Cho hµm sè





 − <

+ ≥

=

6 1; 1

4 ; 1

( )

2

x khix

x x khix f x TÝnh ®¹o hµm cña hµm sè t¹i x0 =1

A -6 B 6 C 5 D 3

3) TÝnh ( )

'

y o cña hµm sè

x

x

y

+

=

1

A -1 B 0 C 2 D 1

4) TÝnh ( )

'

y o cña hµm sè











≠

−

=

=

0

1

0 0

khix

x

Cosx

khix

y

A 2

1

B 4

1

C 2 D - 2

1

5) §¹o hµm cña hµm sè

5

2 3

2

+

− −

=

x

x x

y t¹i x = 0 lµ

A 5

3

− B 5

7

− C 25

7

− D 5

7

2

6) §¹o hµm cña hµm sè 3 7

4 2

y = x − x + t¹i x = -1 lµ

A

5

1

B 5

1

C 5 D 5

7) §¹o hµm cña hµm sè

1

1

2

2

+ +

− +

=

x x

x x

y lµ

A ( )

2 2

2

'

1

2 4 2

+ +

+ −

=

x x

x x

y B ( )

2 2

2

'

1

2

+ +

=

x x

x

y

C ( )

2 2

2

'

1

2 2

+ +

+

=

x x

x

y D ( )

2 2

2

'

1

2 2

+ +

−

=

x x

x

y

8) Ph¬ng tr×nh tiÕp tuyÕn cña ®êng cong 3

y =x t¹i M(1;1) lµ

A y = 3x- 2 B y = -3x+2 C y = 3x+2 D y = -3x- 2

9) Cho H/s 







+ + <

≥

=

1 0

0

( )

2

x x khix

e khix

f x

x

Khi ®ã (0) ?

'

f =

A -1 B 2 C 1 D 4

10) Cho hµm sè f (x) x 3x

2

= + th×

x

f x f

x ∆

− + ∆ − −

∆ →

( 1 ) ( 1)

lim

0

lµ

A -1 B 0 C 1 D 2

11) X¸c ®Þnh a vµ b ®Ó hµm sè

( )









+ + ≥

+ <

=

−

1 0

0

( )

2

ax bx khix

x a e khix

f x

bx

cã ®¹o hµm t¹i x=0

A











=

=

2

1

1

b

a

B











=

=

1

2

1

b

a

C











=

= −

2

1

1

b

a

D











= −

= −

2

1

1

b

a

12) Cho H/s







+ <

− ≥

=

; 1

; 1

( )

2

ax b khix

x x khix f x T×m a vµ b ®Ó H/s cã ®¹o hµm t¹i x= 1

A







= −

=

1

1

b

a

B







=

= −

1

1

b

a

C







= −

= −

1

1

b

a

D







=

=

0

1

b

a

13) §¹o hµm cña hµm sè y = Sinx(1 + Cosx) lµ

A '

y = - Cosx- Cos2x

2

1

B '

y = - Cosx- Cos2x

C '

y = Cosx + Cos2x D '

y = Cosx- Cos2x

14) §¹o hµm cña H/s y = Cosx.Cos3x t¹i 8

π

xo = lµ

A

2

2

−2 − B

2

2

2 + C −8 −2 2 D

4

2

2

1

− −

15) §¹o hµm cña H/s

Sinx Cosx

Sin x Cos x

y

.

2 2

−

= t¹i ®iÓm 6

π

xo = lµ