الأربعاء، 6 مايو، 2015

القواعد الاساسية التفـاضل



التفـــــاضل (1 - 2)


القواعد الأساسية للتفاضل :
1- إذا كانت د(س) = أ حيث أ ثابت فإن :  د/(س) =  = صفر .
2- إذا كانت ص = أس ( أ ثابت ) فإن :  = أ .
3- إذا كانت ص = سن ( حيث ن   ح ) فإن :  = ن سن – 1  .
4- إذا كانت ص = أسن فإن : = أ ن سن – 1  .
المشتقة الأولى للمجموع الجبري لدالتين :
v إذا كانت ع ، ق دالتين في س . وكانت كل منهما قابله للإشتقاق عند س فإن الدالة
ص = ع  ق تكون قابلة للإشتقاق عند س ، يكون :
 =   أي ( ع  ق ) =  
v هذه النتيجة صحيحة لأي عدد منته من الدوال
مشتقة حاصل ضرب دالتين
v إذا كانت ع = ر(س) ، ل = ق(س) ،  ص = ع ل =ر(س) .ق(س) فإن :
 = ع ×  + ل ×

ر(س)
ق(س)
 
ع
ل
 
 مشتقة حاصل قسمة دالتين
         v إذا كانت ع = ر(س) ، ل = ق(س) ص =       =          فإن :
 =
 
                    ل × - ع×
                              ل2
مشتقة دالة الدالة :
v إذا كانت ص = [ د(س) ]ن فإن :       = ن(د(س))ن – 1 ×د/(س) .
مشتقة الدوال المثلثية
v إذا كان ص = جاس فإن  = جتاس.
v إذا كان ص = جتاس فإن  = -جاس .
v إذا كان ص = ظاس فإن  = قا2س .
مسائل منوعة
v جد المشتقة الاولى من المبادئ الأولية للدوال التالية :
(i)                         ص = س5
............................................................................................................      ...................................................................................................... ............................................................................................................      ......................................................................................................
............................................................................................................      ......................................................................................................
............................................................................................................      ......................................................................................................
  5
س3
 
(ii)                      ص =
............................................................................................................      ...................................................................................................... ............................................................................................................      ......................................................................................................
............................................................................................................      ...................................................................................................... ............................................................................................................      ......................................................................................................
(iii)                   ص = س2 – 5
.............................................................. ..............................................      ...................................................................................................... ............................................................................................................      ......................................................................................................
(iv)                   ص = س
............................................................................................................      ...................................................................................................... ............................................................................................................      ...................................................................................................... ............................................................................................................      ...................................................................................................... ............................................................................................................      ......................................................................................................
(v)                      ص = س2 + س + 1
............................................................................................................      ...................................................................................................... ............................................................................................................      ...................................................................................................... ............................................................................................................      ...................................................................................................... ............................................................................................................      ......................................................................................................

v جد  للدوال التالية :
(أ‌)              1/ ص = 6- س3 + جاس
............................................................................................................      ...................................................................................................... ............................................................................................................      ......................................................................................................
2/ ص = 1+ س + جاس
............................................................................................................      ...................................................................................................... ............................................................................................................      ......................................................................................................
3/ ص = س + ظتا2س
............................................................................................................      ...................................................................................................... ............................................................................................................      ......................................................................................................
4/ ص = س3 + 5س2
3
س2
 
............................................................................................................      ...................................................................................................... ............................................................................................................      ......................................................................................................
5/ ص = 4س3 +2س
............................................................................................................      ...................................................................................................... ............................................................................................................      ...................................................................................................... ............................................................................................................      ...................................................................................................... ............................................................................................................      ......................................................................................................



 
       6/ ص =       -        +
س
2
 
............................................................................................................      ...................................................................................................... ............................................................................................................      ...................................................................................................... ............................................................................................................      ...................................................................................................... ............................................................................................................      ......................................................................................................
2
س
 
7/ ص =      +
............................................................................................................      ...................................................................................................... ............................................................................................................      ...................................................................................................... ............................................................................................................      ......................................................................................................
(ب) 1/  ص = س جاس
............................................................................................................      ...................................................................................................... ............................................................................................................      ......................................................................................................
2/ ص= 4س2 ( 3 – 4س )
............................................................................................................      ...................................................................................................... ............................................................................................................      ......................................................................................................
 3/ ص = (2س+5)(س2 + 3س+1)
............................................................................................................      ...................................................................................................... ............................................................................................................      ......................................................................................................
............................................................................................................      ......................................................................................................
4/ ص = س3 جاس
............................................................................................................      ...................................................................................................... ............................................................................................................      ......................................................................................................
5/ ص = جاس جتا2س
............................................................................................................      ...................................................................................................... ............................................................................................................      ......................................................................................................

6/ ص = س2ظا2س
2
س – 2
 
............................................................................................................      ...................................................................................................... ............................................................................................................      ......................................................................................................
(ج) 1/  ص =
............................................................................................................      ...................................................................................................... ............................................................................................................      ...................................................................................................... ............................................................................................................      ...................................................................................................... ............................................................................................................      ......................................................................................................
س2 – س +1
س2 – 3س -2
 
2/ ص =
............................................................................................................      ...................................................................................................... ............................................................................................................      ...................................................................................................... ............................................................................................................      ...................................................................................................... ............................................................................................................      ......................................................................................................
1 - جاس
1 + جتاس
 
3/ ص=
............................................................................................................      ...................................................................................................... ............................................................................................................      ...................................................................................................... ............................................................................................................      ...................................................................................................... ............................................................................................................      ......................................................................................................
(د) 4/ ص = (2س2 – 1) 5
............................................................................................................      ...................................................................................................... ............................................................................................................      ...................................................................................................... ............................................................................................................      ......................................................................................................
2/ ص= (2س – 3 ) 4
............................................................................................................      ...................................................................................................... ............................................................................................................      ...................................................................................................... ............................................................................................................      ......................................................................................................

3/ ص=    2 + 2
............................................................................................................      ...................................................................................................... ............................................................................................................      ...................................................................................................... ............................................................................................................      ...................................................................................................... ............................................................................................................      ........................................................... ...........................................
4/ ص = جا(2-3س) 
............................................................................................................      ...................................................................................................... ............................................................................................................      ......................................................................................................
5/ ص = جتا(2س2 +1)
............................................................................................................      ...................................................................................................... ............................................................................................................      ......................................................................................................  
6/ ص =  جتا2س
                        3
( 3 + 2س + س3)2
 
............................................................................................................      ...................................................................................................... ............................................................................................................      ...................................................................................................... ............................................................................................................      ......................................................................................................

7/ ص =
............................................................................................................      ...................................................................................................... ............................................................................................................      ...................................................................................................... ............................................................................................................      ...................................................................................................... ............................................................................................................      ......................................................................................................
            3
2 – 2 )4
 
8/ ص =

............................................................................................................      ...................................................................................................... ............................................................................................................      ...................................................................................................... ............................................................................................................      ...................................................................................................... ............................................................................................................      ......................................................................................................


        4
س2 + 1
 
9/ ص =  
............................................................................................................      ...................................................................................................... ............................................................................................................      ...................................................................................................... ............................................................................................................      ...................................................................................................... ............................................................................................................      ......................................................................................................
(هـ) 1/ ص = (2س2 +3)( 1-3س)3
        2
(5س – 4 )8
 
............................................................................................................      ...................................................................................................... ............................................................................................................      ...................................................................................................... ............................................................................................................      ...................................................................................................... ............................................................................................................      ...................................................................................................... ............................................................................................................      ......................................................................................................
2/ ص =
............................................................................................................      ...................................................................................................... ............................................................................................................      ...................................................................................................... ............................................................................................................      ...................................................................................................... ............................................................................................................      ...................................................................................................... ............................................................................................................      ......................................................................................................
3/  ص = س    س2 – 4
( س2 – 5 )2
   س+ 2
 
............................................................................................................      ...................................................................................................... ............................................................................................................      ...................................................................................................... ............................................................................................................      ...................................................................................................... ............................................................................................................      ......................................................................................................
4/ ص =
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