La dérivabilité

قﺎــﻘﺘﺷﻻا

La dérivabilité

I -

ﺔﻄﻘﻧ ﻲﻓ قﺎﻘﺘﺷﻻا

ﺪﻴﻬﻤﺗ c: ﺔﻟاﺪﻟا ﺮﺒﺘﻌﻧ ﺔﻓﺮﻌﻤﻟا f

( )

² ــﺑ f x =x

نأ ضﺮﺘﻔﻧ ﻩﺰآﺮﻣ حﻮﺘﻔﻣ لﺎﺠﻣ ﻰﻟإ ﻲﺘﻤﻨﺗ x

0 2

x =

ﻊﻀﻧو x= +2 h

ﺎﻨﻳﺪﻟ

( ) (

²

)

^{4 4 2} f x = f +h = + h h+

0 h h2

4 4h+

ﺖﻧﺎآ اذإ نﺈﻓ

ﺔﻠﻤﻬﻣ نﻮﻜﺗ .

دﺪﻌﻟا نا ﺮﺒﺘﻌﻧ ﻪﻨﻣو دﺪﻌﻠﻟ ﺔﺑﺮﻘﻣ ﺔﻤﻴﻗ ﻮه

(

²

)

f +h .

لﺎﺜﻣ :

(

^{2, 01}

) (

^{2 0, 01}

)

^{4 0, 04 4, 04}

f = f + = + =

ﺔﻈﺣﻼﻣ c ﺎﻨﻳﺪﻟ

( )

^{2 4}

f =

(

²

) ( )

^{2 4 2} نذإ f + −h f = h h+

ﻞﺟأ ﻦﻣو ﺪﺠﻧ h≠0

(

²

) ( )

²

f h f 4 h h + −

= +

ﻪﻨﻣو

( ) ( )

0

2 2

m 4

h

f h f

→ h

+ −

li =

d دﺪﻌﻟا نأ ﺎﻧﺮﺒﺘﻋا 4 4h+

ﻟ ﺐﻳﺮﻘﺗ ﻮه

( )

2+h ² دﺪﻌﻠ

اﺬهو نأ ﻲﻨﻌﻳ

( )

4 4+ x−2

2 دﺪﻌﻠﻟ ﺐﻳﺮﻘﺗ ﻮه راﻮﺠﺑ x

2

ﻢﻴﻘﺘﺴﻤﻟا ﺊﺸﻧأ

: 4 4

D y= x− ﻰﻨﺤﻨﻤﻟاو

( )

^A

؟ ﻆـﺣﻼﺗ اذﺎـــــــــﻣ

ﺔﺻﻼﺧ : ﺘﺴﻤﻟا

( )

^ﻴﻘ

( )

^{A D ﻢ}

ﺔﻄﻘﻨﻟا ﻲﻓ ﻰﻨﺤﻨﻤﻠﻟ سﺎﻤﻣ

( )

^{2, 4}

A

4 4

x6 x− f

.

ﺔﻟاﺪﻟا ﻰﻤﺴﺗ

ﺔﺳﺎﻤﻤﻟا ﺔﻴﻔﻟﺂﺘﻟا ﺔﻟاﺪﻟا ﺔﻟاﺪﻠﻟ

لﻮﺼﻓﻷا تاذ ﺔﻄﻘﻨﻟا ﻲﻓ 2

.

ﺔﻄﻘﻧ ﻲﻓ قﺎﻘﺘﺷﻻا

ﻈﺣﻻ

ﺎﻨ بﺎﺴﺣ نأ ﺔﻘﺑﺎﺴﻟا ﺔﻠﺜﻣﻷا ﻲﻓ

(

0

) ( )

0

0

lim

h

f x h f x

→ h

+ −

0راﻮﺠﺑ ﺔﻴﻔﻟﺂﺗ ﺔﻟاﺪﺑ f ﺐﻳﺮﻘﺘﻟ ﺎﻧﺪﻋﺎﺴﻳ .x

ﻒﻳﺮﻌﺗ ﻦﻜﺘﻟ ﻩﺰآﺮﻣ حﻮﺘﻔﻣ لﺎﺠﻣ ﻰﻠﻋ ﺔﻓﺮﻌﻣ ﺔﻟاد .

x₀ f

ﻲﻘﻴﻘﺣ دﺪﻋ ﺪﺟو اذإ ﻂﻘﻓو اذإ ﺚﻴﺤﺑ l

x₀ ﻲﻓ قﺎﻘﺘﺷﻼﻠﻟ ﺔﻠﺑﺎﻗ f ﺔﻟاﺪﻟا نأ لﻮﻘﻧ Ž

(

0

) ( )

0

m0 h

f x h f x h l

→

+ −

=

0 f li

دﺪﻌﻟا Ž دﺪﻌﻟا ﻰﻤﺴﻳ l ﻖﺘﺸﻤﻟا

ﺔﻟاﺪﻠﻟ ﺔﻄﻘﻨﻟا ﻲﻓ

.x

ﺐﺘﻜﻧو

( )

0

' x

= l f

ﺔﻈﺣﻼﻣ ﺎﻨﻌﺿو اذإ x=x0+h

ن

( ) ( ) ( ) ( )

ﺈﻓ

0

0 0 0

0 0

lim lim

h x x

f x h f x f x f x

h x x l

→ →

+ − −

= =

−

( )

( ) (

0 0

) (

0

) ( )

نذإ f x = f x +l x−x + −x x ϕ x

( )

ﻊﻣ

0

limϕ x 0

→

x x =

ﻒﻳﺮﻌﺗ : ﺔﻟاﺪﻟا

( ) (

0 0

)

x6 f x +l x−x

0ﺔﻄﻘﻨﻟا ﻲﻓ f ﺔﻟاﺪﻠﻟ ﺔﺳﺎﻤﻤﻟا ﺔﻴﻔﻟﺂﺘﻟا ﺔﻟاﺪﻟا ﻰﻤﺴﺗ .x

ﺔﻠﺜﻣأ : ﺔﻟاﺪﻟا ﻞه ﻲﻓ قﺎﻘﺘﺷﻼﻟ ﺔﻠﺑﺎﻗ f

x0

؟ ﺔﻴﻟﺎﺘﻟا تﻻﺎﺤﻟا ﻲﻓ

a

( )

² ( f x =x +x

0 1

x =

b

( )

¹ ( f x = x+

0 0

x =

c

( )

₂¹ ( f x 1

= x +

0 1

x =

d

( )

( f x = x

0 0

x =

ﺔﻴﺻﺎﺧ : ﻲﻓ قﺎﻘﺘﺷﻼﻟ ﺔﻠﺑﺎﻗ ﺔﻟاد ﻞآ ﻲﻓ ﺔﻠﺼﺘﻣ نﻮﻜﺗ

x0 x₀

.

ﺔﻈﺣﻼﻣ : ﺢﻴﺤﺻ ﺮﻴﻏ ﺔﻴﺻﺎﺨﻟا ﻩﺬه ﺲﻜﻋ .

II - ﻦﻴﻤﻴﻟا ﻰﻠﻋ قﺎﻘﺘﺷﻹا –

رﺎﺴﻴﻟا ﻰﻠﻋ قﺎﻘﺘﺷﻹا

1 ( ﺪﻴﻬﻤﺗ : ﻦﻜﺘﻟ ﻰﻠﻋ ﺔﻓﺮﻌﻣ ﺔﻳدﺪﻋ ﺔﻟاد ﻲﻠﻳ ﺎﻤﺑ

: \ f

( )

2 1, 0

1 , 0

1

x ax x

x x

x

⎧ + + ≤

= ⎨⎪

⎪ +⎩ ; a

f f

f ﺚﻴﺣ مﻮﻠﻌﻣ ﻲﻘﻴﻘﺣ دﺪﻋ .

1 ( نأ ﻦﻴﺑ ﻲﻓ ﺔﻠﺼﺘﻣ

0

2 ( ﺔﻤﻴﻗ دﺪﺣ نﻮﻜﺗ ﻲﻜﻟ a

ﻲﻓ قﺎﻘﺘﺷﻺﻟ ﺔﻠﺑﺎﻗ 0

2 ( ﻒﻳرﺎﻌﺗ

a - ﻦﻴﻤﻴﻟا ﻰﻠﻋ قﺎﻘﺘﺷﻻا

b - رﺎﺴﻴﻟا ﻰﻠﻋ قﺎﻘﺘﺷﻻا

3 ( ﻬﻟا ﻞﻳوﺄﺘﻟا ﻲﺳﺪﻨ

"

ﻰﻨﺤﻨﻤﻠﻟ سﺎﻤﻤﻟا ﻒﺼﻧ ﺔﻟدﺎﻌﻣ ﺔﻄﻘﻨﻟا ﻲﻓ A

( ( ) )

0 0, 0

M x f x

( )

^A

"

ﻜﻴﻟ Ž ﺔﻟاﺪﻠﻟ ﻞﺜﻤﻤﻟا ﻰﻨﺤﻨﻤﻟا

و f ﻦ

( ( ) )

0 0, 0

M x f x A

0 f ﻦﻣ ﺔﻄﻘﻧ

( )

.

ﺖﻧﺎآ اذإ ﻦﻴﻤﻴﻟا ﻰﻠﻋ قﺎﻘﺘﺷﻺﻟ ﺔﻠﺑﺎﻗ

) رﺎﺴﻴﻟا ﻰﻠﻋ وأ (

ﻲﻓ x

ﻰﻨﺤﻨﻤﻟا نﺈﻓ ﺎﻬﻟﻮﺼﻓأ ﻲﺘﻟا ﺔﻄﻘﻨﻟا ﻲﻓ سﺎﻤﻣ ﻒﺼﻧ ﻞﺒﻘﻳ

ﻮه ﻪﺟﻮﻤﻟا ﻞﻣﺎﻌﻤﻟا نﻮﻜﻳو

( )

0

'_d f x )

( )

0 وأ '_g

f x x0

( )

^A

(

ﺜﻣ ــ لﺎ :

( )

( )

2

2

1, 0 1, 0

x x

f x x x x

⎧f x = +

⎪⎨

= − + ≤

⎪⎩

;

ﺔﻈﺣﻼﻣ (*) ﺖﻧﺎآ اذإ ﺎﻘﺘﺷﻺﻟ ﺔﻠﺑﺎﻗ f

نﺈﻓ ق

( )

0

( )

0

'_g '_d

f x = f x

0 f

(*) ﺖﻧﺎآ اذإ ﻲﻓ ﻦﻴﻤﻴﻟا ﻰﻠﻋ قﺎﻘﺘﺷﻺﻟ ﺔﻠﺑﺎﻗ

x

ﻟ ﺔﻠﺑﺎﻗو ﻲﻓ رﺎﺴﻴﻟا ﻰﻠﻋ قﺎﻘﺘﺷﻺ

x0

آو نﺎ

( )

0

( )

0

'_g '_d

f x = f x

0 f

نﺈﻓ ﻲﻓ قﺎﻘﺘﺷﻺﻟ ﺔﻠﺑﺎﻗ

x

ﺎﻨﻳﺪﻟو

( )

0

( )

0

( )

0

' '_g '_d

f x = f x = f x

III -

ﺔﻘﺘﺸﻤﻟا ﺔﻟاﺪﻟا

[ ]

^{a b,}

I = حﻮﺘﻔﻣ لﺎﺠﻣ ﻰﻠﻋ قﺎﻘﺘﺷﻹا ﺔﻴﻠﺑﺎﻗ cﻒﻳﺮﻌﺗ:ﻒﻳرﺎﻌﺗ (1 ﻒﻳﺮﻌﺗ

d ﻖﻠﻐﻣ لﺎﺠﻣ ﻰﻠﻋ قﺎﻘﺘﺷﻹا ﺔﻴﻠﺑﺎﻗ

[ ]

^{a b,}

f

ﻒﻳﺮﻌﺗ e

ﻦﻜﺘﻟ لﺎﺠﻣ ﻰﻠﻋ قﺎﻘﺘﺷﻺﻟ ﺔﻠﺑﺎﻗ ﺔﻟاد I

ﻰﻠﻋ ﺔﻓﺮﻌﻤﻟا ﺔﻟاﺪﻟا ﻲﻠﻳ ﺎﻤﺑ I

:

f ' :I →\

( )

' x6 f x f

ﺔﻟاﺪﻟا ﻰﻤﺴﺗ ﺔﻘﺘﺸﻤﻟا

ﺔﻟاﺪﻠﻟ لﺎﺠﻤﻟا ﻰﻠﻋ

ــﺑ ﺎﻬﻟ ﺰﻣﺮﻧو I f '

ﺔﻠﺜﻣأ :

1

(

ﺔﻘﺘﺸﻣ ﺔﻟاﺪﻟا

( )

f x =k

2 ( ﺔﻟاﺪﻟا ﺔﻘﺘﺸﻣ

( )

f x =x

3 ( ﺔﻟاﺪﻟا ﺔﻘﺘﺸﻣ

( )

² f x =x

4 ( ﺔﻟاﺪﻟا ﺔﻘﺘﺸﻣ

( )

ⁿ n∈` f x =x

ﺘﻨﺘﺳا ــ جﺎ .

2 ( ﻘﻟا لاوﺪﻟا ﻰﻠﻋ تﺎﻴﻠﻤﻌﻟا قﺎﻘﺘﺷﻺﻟ ﺔﻠﺑﺎ

(

f +g

) ( )

' x0 = (1 2 (

( ) ( )

kf ' x0 =

(

f g.

) ( )

' x0 = (3 4 (

( )

0

1 ' x

⎛ ⎞

g =

⎜ ⎟⎝ ⎠

5 (

( )

0

f ' g x

⎛ ⎞ =

⎜ ⎟⎝ ⎠

تﺎـﻘﻴﺒﻄﺗ

1 ( ﺘﺸﻣ ﺔﻳدوﺪﺤﻟا لاوﺪﻟا ﺾﻌﺑ ﺔﻘ .

1

xn ﺔﻟاﺪﻟا ﺔﻘﺘﺸﻣ (2 ﺔــﻠـﺜﻣأ

جﺎﺘﻨﺘﺳا

3 ( ﺔﻟاﺪﻟا ﺔﻘﺘﺸﻣ

( )

^{ax b} cx d f x = + 0 +

c≠

ﺔــﻠـﺜﻣأ

جﺎﺘﻨﺘﺳا

4 ( ﺟ ﺔﻟاد ﺔﻘﺘﺸﻣ ﺬ

ﺔـﻳر

3 ( ﺔﻳدﺎﻴﺘﻋﻻا ﺔﻴﺜﻠﺜﻤﻟا لاوﺪﻟا ﺔﻘﺘﺸﻣ

ﻴآﺬﺗ ــ ﺮ :

sin sin 2 cos sin

2 2

p q p q

p− q= + −

cos cos 2 sin sin

2 2

p q p q

p− q= − + −

1.3 . اﺪﻟا ﺔﻘﺘﺸﻣ

( )

^sin ﺔﻟ f x = x

ﺔﻟاﺪﻟا ﺔﻘﺘﺸﻣ .2.3

( )

^cos

f x = x

( )

^tan ﺔ

f x = x ﻟاﺪﻟا ﺔﻘﺘﺸﻣ .3.3

ﺔﻨهﺮﺒﻣ

( )

^{un ' .} : قﺎﻘﺘﺷﻺﻟ ﺔﻠﺑﺎﻗ ﺔﻟاد

.

u =nuⁿ⁻¹u'

( )

ﻘﻴﺒﻄﺗ ــ تﺎ ﺔﻴﻟﺎﺘﻟا لاوﺪﻟا ﺔﻘﺘﺸﻣ دﺪﺣ :

1 (

(

^{2 3}

)

³

f x = x − x

2

( )

^sin2 (

f x = x

3

( ) (

^{2 1 tan}

)

³ (

f x = x + x

4

( )

^{cos sin} (

f x = x− x

( )

5 (

2 cos sin

cos

x x

f x x

= −

( )

6 (

tan 1 cos f x x

x

= −

f

ﺔﻨهﺮﺒﻣ : ﻦﻜﺘﻟ و قﺎﻘﺘﺷﻺﻟ ﺔﻠﺑﺎﻗ ﺔﻟاد ــﺑ ﺔﻓﺮﻌﻣ ﺔﻟاد g

^{g x}

( )

^{= f ax b}

(

⁺

)

ﺔﻟاﺪﻟا ﺔﻘﺘﺸﻣ ه

ــ ﺔﻟاﺪﻟا ﻲ g

( ) ( )

' '

g x =af ax b+

ﻠﺜﻣأ ــ ﺔ

( )

: '

⇒ f

( )

^cos

( )

f x = ax b+ x =?

( )

'

( )

^sin

( )

f

f x = ωx+ϕ ⇒ x =?

( )

'

f ?

⇒

( )

^tan

( )

f x = ax b+ x =

( )

' ?

f x =

⇒

( )

^{cos 2 sin} 3 f x = x− x

4 ( ﺔﻳرﺬﺟﻼﻟا لاوﺪﻟا ﺔﻘﺘﺸﻣ

1.4 . ﺔﻟاﺪﻟا ﺔﻘﺘﺸﻣ

( )

f x = x

ﺔ

( ) ( )

ﻟاﺪﻟا ﺔﻘﺘﺸﻣ ^.2.4 f x = u x

تﺎـــﻘﻴﺒﻄﺗ

ﺔﻘﺘﺸﻤﻟا لاوﺪﻟا لوﺪﺟ ﺔﻳدﺎﻴﺘﻋﻻا لاوﺪﻠﻟ

.

IV -

ﻲﺳﺪﻨﻬﻟا ﻞﻳوﺄﺘﻟا

M0

و نﺎﺘﻔﻠﺘﺨﻣ نﺎﺘﻄﻘﻧ M ﻦﻣ

. A

x0

لﻮﺼﻓأ M0

.

x0+h لﻮﺼﻓأ .M

ﻢﻴﻘﺘﺴﻤﻠﻟ ﻪﺟﻮﻤﻟا ﻞﻣﺎﻌﻤﻟا

(

M M0

)

ﻮه

f x

(

0 h

)

f x

( )

0

h + −

ﺖﺑﺮﺘﻗا ﺎﻤﻠآ ﻪﻧأ ﻆﺣﻼﻧ ﻦﻣ M

M0

(

M M0

)

نﺈﻓ T

( )

ﻦﻣ بﺮﺘﻘﺗ .

نﺈﻓ ﻪﻨﻣو ﻢﻴﻘﺘﺴﻤﻠﻟ ﻪﺟﻮﻤﻟا ﻞﻣﺎﻌﻤﻟا

ﻮهT

( ) ( )

0

0 0

m ' 0

h

f x h f x

li f x

h

→

+ −

=

ﻒﻳﺮﻌﺗ ﻦﻜﺘﻟ

ﻲﻓ قﺎﻘﺘﺷﻼﻟ ﺔﻠﺑﺎﻗ ﺔﻟاد f x0

ﺔﻟاﺪﻠﻟ ﻞﺜﻤﻤﻟا ﻰﻨﺤﻨﻤﻟا

. ^f

( )

^A و

0ﺔﻄﻘﻨﻟا ﻲﻓ f ﺔﻟاﺪﻠﻟ ﺔﺳﺎﻤﻤﻟا ﺔﻴﻔﻟﺂﺘﻟا ﺔﻟاﺪﻠﻟ ﻞﺜﻤﻤﻟا ﻢﻴﻘﺘﺴﻤﻟا ﻰﻤﺴﻳ x

سﺎﻤﻣ ﻨﺤﻨﻤﻟا لﻮﺼﻓﻷا تاذ ﺔﻄﻘﻨﻟا ﻲﻓ

( )

^A ﻰ x0

.

ﻲه سﺎﻤﻤﻟا ﺔﻟدﺎﻌﻣو

:

( )(

0 0

) ( )

0

'

y= f x x−x + f x