gxf'1x1f1 =-+ 2x1112x1fxf12x1limlimlimx1x1x1 -+-+-+==--- gxf'xxxfx =-+ f'13 = f11 = fx2xx =- f'xA =

سرد : ق ا م وا ى

زإ ذ ا : !"ا #$

ا ر %&ا '()

*(ز+&ا

Chorfi_mouhsine@yahoo.fr

ــI د- 'اد ق ا './0 :

1(!+2 :

**

اا نإ ل

ق f

0 ا

د و اذإ x ! A

:

( ) ( )

0

0 x x

0

f x f x

lim A

x x

→

− =

وأ −

(

⁰

) ( )

⁰

h 0

f x h f x

lim A

→ h

+ −

=

**

د$ا %&'ا د$ا (')* A ا

ا f x0

+, و

( )

⁰

f ' x =A .

ل4 : قا سر ا f

! 1

( )

²

f x =2x −x .

( ) ( )

²

x 1 x 2

f x f 1 2x x 1

lim lim

x 1 x 1

→ →−

− = − −

− −

)

( )

ن1 f 1 =1 (

( ) ( )( )

x 1 x 1 x 1

2 x 1 x 1

2x 1 x 1

lim 2 lim lim 2x 1 3

x 1 x 1

→ → →

 

+ −

  + −

 

= = = + =

− −

) 3'$ا و 4''ا ب) $

(

اا نذإ

ا ق6 f

* و 1

( )

f ' 1 =3 .

I ــI د- 'ا- '&&ا '.5.6 ا 'ا-ا :

1(!+2 :

7, 84آ:; ح=; ل>; ( :$; *د اد f x0

.

اذإ آ اا ?

ق f

x0

ا @''ا =Aا اا نB f

x0

اا ه

( ) ( )(

0 0

) ( )

0

g x =f ' x x−x +f x .

ل4 : اا :D$

3,&ا ( :$'ا f

( )

¹ : f x = x 1 . +

1 ا %&'ا د$ا د! ــ f

1 .

( ) ( )

( )

( )

x 1 x 1 x 1

2 x 1

1 1

2 x 1

f x f 1 x 1 2

lim lim lim

x 1 x 1 x 1

→ → →

− +

− +

− = + =

− − −

( ) ( ) ^{( )} ( ) ^{( )} ( )

x 1 x 1 x 1

2 x 1 1 x x 1

lim lim lim

2 x 1 x 1 2 x 1 x 1 2 x 1 x 1

→ → →

− − = − = − −

− + − + − +

( )( )

( ) ( )( ) ^{( )} ( )( ) ( )( )

x 1 x 1 x 1

x 1 x 1 x 1 1 1

lim lim lim

2 x 1 x 1 x 1 2 x 1 x 1 x 1 2 x 1 x 1 8

→ → →

− + −

= − = − = − = −

− + + − + + + +

2 ل @''ا =Aا اا د! ــ f

1 .

( ) ( )( ) ( )

g x =f ' 1 x 1− +f 1

H;و

( )

¹

( )

¹

g x x 1

8 2

= − − +

3 د$ :; ' د ــ 1

1.0001 1+ .

نأ '

( )

¹

( )

¹

g x x 1

8 2

= − − + اا نB

ا +*:J g د$ا را> f

1 H;و

( )

¹

( )

¹

f x x 1

8 2

− − +

. ≃

KL h= −x 1 نذإ

x= +h 1

( )

^{1 1} و

f h 1 h

8 2

+ ≃ − + .

* : 0.499988

( ) ( ) ( )

=

1 1 1

f 1.0001 f 0.0001 1 0.0001

8 2

1.0001 1

= = + ≈ − +

نأ NO + h=0.0001

.

fr . yahoo

@ mouhsine _

Chorfi

I ــII ر.ا و #.&.ا ق ا :

1 ــ #.&.ا ق ا :

1(!+2 :

7,

=; ل>; ( :$; اد f ح

ع 7;

0 0

x , x +a

 

 

! a>0 .

نإ ل

7'ا ( ق f

x0

اذإ ا ?آ

( ) ( )

⁰

0

f x f x x x

− *Q −

7'ا ( l x0

ـ Q 4;: و

( )

d 0

f ' x .

د$ا ا %&'ا د$ا (')* l ا ( f

0 7' . x

+,و

( ) ( ) ( )

:

0

0

d 0

x x

0

f x f x

lim f ' x

x x

→ +

− =

−

2 ق ا ــ ر.ا

:

1(!+2 :

7, ع 7; ح=; ل>; ( :$; اد f

0 0

x −a, x

 

 

! a>0 .

نإ ل

ر)ا ( ق f

x0

ا ?آ اذإ

( ) ( )

⁰

0

f x f x x x

− *Q −

ر)ا ( l x0

ـ Q 4;: و

( )

g 0

f ' x .

د$ا ا %&'ا د$ا (')* l ر)ا ( f

x0

. +,و

( ) ( ) ( )

:

0

0

g 0

x x

0

f x f x

lim f ' x

x x

→ −

− =

. −

3 '.:; ــ :

ن,J

ق f

x0

ذإ R و اذإ ?آ ا

ر)ا ( و 7'ا ( ق f

x0

د$ا يو)* 7'ا ( %&'ا د$ا و

ر)ا ( %&'ا .

($' :

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

0 0 0

0 0 0

0 0

x x x x x x

0 0 0

f x f x f x f x f x f x

lim f ' x lim lim f ' x

x x ⁺ x x ⁻ x x

→ → →

− − −

= ⇔ = =

− − −

ل4 : اا قا سر

( )

^{2 x}

f x = x 1 −

0 .

**

7'* ( قUا سر 0

:

( ) ( )

( ) ( )

x 0 x 0 x 0 x 0 x 0

2x 2x

f x f 0 x 1 0 x 1 2x 2

lim lim lim lim lim 2

x 0 x 0 x x x 1 x 1

+ + + + +

→ → → → →

− = − − = − = = = −

− − − −

نإ ل

7'* ( ق f

0 .

**

ر)* ( قUا سر 0

:

( ) ( )

( ) ( )

x 0 x 0 x 0 x 0 x 0

2x 2x

f x f 0 x 1 0 x 1 2x 2

lim lim lim lim lim 2

x 0 x 0 x x x 1 x 1

− − − − −

→ → → → →

− − −

− = − = − = − = − =

− − − −

نإ ل

ر)* ( ق f

0 .

**

نB ر)ا ( %&'ا د$ا VW* 7'ا ( %&'ا د$ا نأ '

ق :X f

0 .

ــIV ق < -=>ا ?(و6 ا :

( )

Cf

1 س&&ا ــ :

1(!+2 :

7, 84آ:; ح=; ل>; ( :$; *د اد f x0

.

( )( ) ( )

y =f ' x0 x−x0 +f x0

اا ?آ اذإ

ق f

x0

( )

Cf نB

@'; 3D*

( )

0

f x

QYأ ا ا x0

Hد$;

( )(

0 0

) ( )

0 : y =f ' x x−x +f x .

( )

⁰ د$ا f ' x ـ س'' H'ا 3;$'ا ه

( )

^Cf

QYأ ا ا x0

. x0

ل4 :

1 ا %&'ا د$ا ــ

( )

²

f x =2x −x د$ا

ه 1 . 3

ـ س''ا د$;

( )

^Cf

ا ا QYأ

ه 1

( )( ) ( )

: y =f ' 1 x 1− +f 1

( )

y=3 x 1− +1

fr . yahoo

@ mouhsine _

Chorfi

y=3x−2

2 ا %&'ا د$ا ــ

( )

²

f x =x +2x د$ا

−1 ه . 0

; ـ س''ا د$

( )

^Cf

QYأ ا ا

−1 ه

( )( ) ( )

: y =f ' −1 x 1+ + −f 1

y= −1

نإ ل

( )

^Cf

Hد$; أ @'; 3D*

y = −1

.

2 ــ 1@

س&&ا :

1(!+2 1 :

( )

0

f x

7, د اد f ع 7; ل>; ( :$; *

0 0

x , x +a

 

 

.

اا ?آ اذإ

ق f

7'* ( x0

( )

Cf نB 3D*

VY س';

x0

QYأ ا ا x0

Hد$;

( )( ) ( )

:

d 0 0 0

y =f ' x x−x +f x .

1(!+2 2 :

7, ع 7; ل>; ( :$; *د اد f

0 0

x −a, x

 

 

.

( )

⁰

f x

اا ?آ اذإ

ر)* ( ق f

x0

( )

^Cf نB س'; VY 3D*

QYأ ا ا x0

Hد$;

( )( ) ( )

:

g 0 0 0

y =f ' x x−x +f x .

x0

'ABC :

* اا ?آ اذإ

ق f

7'* ( x0

) ر)* ( وأ x0

( نآ و

( ) ( )

0

0 x x

0

f x f x

lim 0

x x

→ +

− =

) −

( ) ( )

وأ

0

0 x x

0

f x f x

lim 0

x x

→ −

− =

( −

نإ ل

( )

^Cf

7'* ( أ س'; VY 3D*

x0

) ر)* ( وأ x0

. (

* اا ?آ اذإ

ق :X f

7'* ( x0

) ر)* ( وأ x0

( نآ و

( ) ( )

0

0 x x

0

f x f x lim^{→ +} x x

− = ∞

) − وأ

( ) ( )

0

0 x x

0

f x f x lim^{→ −} x x

− = ∞

( − نإ ل

( )

^Cf

7'* ( يد' س'; VY 3D*

x0

) ( وأ

0 ر)*

. ( x

ل4 :

**

اا قإ سر

( )

²

f x =x −x 7'* (

@ه >ا لوA و 1 .

( ) ( )

^{2 2}

( )

x 1 x 1 x 1 x 1 x 1

f x f 1 x x 0 x x x x 1

lim lim lim lim lim x 1

x 1 x 1 x 1 x 1

+ + + + +

→ → → → →

− = − − = − = − = =

− − − −

اا

7'* ( ق f

1 .

( )

Cf

7'* ( س'; VY 3D*

1 Hد$;

( )( ) ( )

: y =f ' 1 x 1− +f 1

y = −x 1

**

اا قإ سر

( )

²

f x =x −x ر)* (

1 @ه >ا لوA و 2 .

( )

^{2 2}

1 1 1

x x x

2 2 2

1 1 4x 4x 1

f x f x x

2 4 4

lim lim lim

1 1 4x 2

x x

2 2 4

− − −

→ → →

  − +

−   − +

  = = −

− −

( )

( )

2 2

1 1 1

x x x

2 2 2

4x 4x 1 2x 1 2x 1

lim lim lim 0

4x 2 2 2x 1 2

− − −

→ → →

− + − −

= = = =

− −

اا

( ق f

1 ر)*

. 2

( )

^Cf

س'; VY 3D*

أ ( 1ر)*

Hد$; 2

1 1 1 :

y f ' x f

2 2 2

     

=    − +  

     

Chorfi_mouhsine@yahoo.fr

y 1 4

= −

ل4 :

**

اا قإ سر

( )

f x = x 1− 7'* (

@ه >ا لوA و 1 .

( ) ( )

x 1 x 1 x 1

f x f 1 x 1 1 1

lim lim lim

x 1 x 1 x 1 0

+ + + +

→ → →

− = − = = = +∞

− − −

اا

7'* ( ق :X f

1 .

( )

Cf

7'* ( يد' س'; VY 3D*

1 Hد$;

: x =1

ــV ' "&ا 'ا-ا :

1 1(ر+2 ــ :

**

اا نإ ل ح='ا ل>'ا ( ق f

?آ اذإ I

7; 3آ ق f

. I

**

اا نإ ل ل>'ا ( ق f

ab

  ?آ اذإ ل>'ا ( ق f

 ab 7'* ( و   ر)* و a

. b

**

اا 7, ل>'ا ( ق f

.I

:Y 3آ R:J ا اا 7; x

د$ I

( )

f ' x ا &'ا اا (')J ـ Q 4;: f

. f '

**

اا 7, ل>'ا ( ق f

.I

اا ?آ اذإ f '

ل>'ا ( ق ــ Q 4;: و Zا &'ا اا (')J &'ا Qاد نBI

. f ''

2 '(د. ا لاو-ا ' " ــ :

^{f x}

( )

^=a

Df =ℝ

( )

f ' x =0

f ' D =ℝ

( )

f x =x

f D =ℝ

( )

f ' x =1

f ' D =ℝ

( )

^{n *}

f x =x (n∈ℕ )

f D =ℝ

( )

^{n 1} f ' x =nx ⁻

f ' D =ℝ

( )

f x = x

Df =ℝ⁺

( )

¹ f ' x

= x

*

Df ' =ℝ⁺

( )

f x =sin x

f D =ℝ

( )

f ' x =cos x

f ' D =ℝ

( )

f x =cos x

f D =ℝ

( )

f ' x = −sin x

f ' D =ℝ

3 ــ ' "&ا لاو-ا ت.&+ا :

'.:;

: 7, و f

ق 7 7اد g

x0

λ و د .

**

'ا-ا f +g ق < '/0 x0

=(- و

(

^{f +g '}

)

^{= +f ' g '} : .

ل4 :

*

( )

^{3 2}

f x = x +x + x

( ) ( ) ( )

^{3 ' 2 '}

( )

^{' 2 1}

f ' x x x x 3x 2x

= + + = + +2 x

*

( )

⁵

f x =sin x+cos x+x +4

( ) ( ) (

^'

)

^'

( )

^{5 '}

^{( )}

^{' 4}

f ' x = sin x + cos x + x + 4 =cos x−sin x +5x

**

'ا-ا λf ق < '/0 x0

=(- و

( )

^{λf '= λf '} : .

ل4 : *

( )

f x =5 sin x

( ) ( )

^'

( )

^'

f ' x = 5 sin x =5 sin x =5 cos x

*

( )

f x = −3 x

( ) ( )

^'

( )

^{' 1 3}

f ' x 3 x 3 x 3

2 x 2 x

= − = − = − × = −

**

'ا-ا f×g ق < '/0 x0

=(- و

(

^{f×g '}

)

^{= × + ×f ' g f g'} : .

ل4 : *

( ) (

²

)

f x = x +x x Chorfi_mouhsine@yahoo.fr

( ) ( (

²

) )

^'

⁽

²

⁾

^'

⁽

²

⁾ ( )

^'

f ' x = x +x x = x +x × x + x +x x

(

^{2x 1}

)

^x

(

^{x2 x}

)

_{2 x}¹

= + × + + ×

(

^{2x 1}

)

^{x 2 x}

(

^{x2 x}

)

2 x

+ × × + +

=

2 2 2

4x 2x x x 5x 3x

2 x 2 x

+ + + +

= =

*

( )

³

f x =x sin x

( ) (

³

) ( )

^{' 3 ' 3}

^{( )}

^{' 2 3}

f ' x = x sin x = x ×sin x+x × sin x =3x sin x+x cos x

**

نآ اذإ

( )

0

g' x ≠0 'ا-ا

f '

g

   ق < '/0   x0

=(- و :

'

2

f f ' g f g '

g g

  = × − ×

   .

ل4 : *

( )

²2

x x

f x x 1

= + +

( ) ( ) ( ) ( )( )

( )

' '

' 2 2 2 2

2

2 2 2

x x x 1 x x x 1

x x

f ' x

x 1 x 1

+ + − + +

 + 

=  =

 +  +

( ) ( ) ( )

( )

2 2

2 2

2x 1 x 1 x x 2x x 1

+ + − +

= +

( ) ( )

3 2 3 2 2

2 2

2 2

2x 2x x 1 2x 2x x 2x 1

x 1 x 1

+ + + − − − + +

= =

+ +

*

( )

2

f x 2 x

x 1

= −

( ) ( ) ^{( )} ( ) ^{( )}

( )

' '

' 2 2

2 2

2

2 x x 1 2 x x 1

f ' x 2 x

x 1 x 1

− − −

 

= −  = −

( )

( )

2

2 2

2 1 x 1 2 x 2x

2 x

x 1

× × − − ×

= −

( ) ( ) ( ) ( )

2 2 2

2 2 2

2 2 2 2

2 2 2 2

x 1 x 1 4x

4x x x 1 4x 3x 1

x x x

x 1 x 1 x x 1 x x 1

− − − −

− − − −

= = = =

− − − −

4 ــ > " ةرإ و 'اد '/2ر :

'=ه!L : 7, ا &'ا اا f '

ل>'ا ( f . I

**

?آ اذإ ( ;$; f '

نB I ( [ ن,J f . I

**

?آ اذإ ( D; f '

نB I ( **ا4J ن,J f . I

**

?آ اذإ ( D@ f '

نB I ( YJ ن,J f . I

ل4 :

ـــــ ا &'ا اا د!

QJا:\J لو KL و f .

**

1

( )

^{3 2} **

f x =x −2x +4

**

&'ا اا

( ) (

^{3 2}

)

^{' 2} : f ' x = x −2x +4 =3x −4x Chorfi_mouhsine@yahoo.fr

**

اا ةرإ : f '

دو!ا [6[ 4'; +)!

3x2 −4x 'ه 7=W; 7 ( 3Y!

: 0 4و . 3

+∞

4

3 0

−∞

x

+ ــــ

+ 3x2 −4x

**

اا تا:\J لو . f

+∞

4

3 0

−∞

x

+ ــــ

+

( )

f ' x

+∞

4

76

27

−∞

( )

f x

**

2 **

( )

²2

x 4

f x x 1

= − +

**

&'ا اا

( ) ( ) ( ) ( )( )

:

( ) ( ) ( )

( )

' '

' 2 2 2 2 2 2

2

2 2 2

2 2

x 4 x 1 x 4 x 1 2x x 1 x 4 2x

x 4

f ' x

x 1 x 1 x 1

− + − − + + − −

 − 

=  = =

 +  + +

( ) ( )

3 3

2 2

2 2

2x 2x 2x 8x 10x

x 1 x 1

+ − +

= =

+ +

**

اا تا:\J لو . f

+∞

0

−∞

x

+ ـــــ

f '

1

1

f

−4

4 ــ ل ق < '/0 'اد فا!N : I

'.:;

:

7, و ح=; ل>; I x0

ل>'ا 7; ا:Y . I

اا ?آاذإ ــ

ق f

x0

ا:; 3DJ و x0

نB

( )

0 : f ' x =0 .

نآ اذإ ــ

( )

0

f ' x =0 ?آ و

إ :\J f ' را> QJر x0

نB ا:; 3DJ f x0

.

Chorfi_mouhsine@yahoo.fr