Lecture

1. lecture 4
Prepared by:
Assist. Lect. Hiba Abdul –Kareem Assist. Lect. Azam Isam

2.    
0 is a constant
d
c c
dx

   
1
is a real number
n n
d
x nx n
dx


     
( ) ( ) is a constant
d d
cf x c f x c
dx dx

       
( ) ( )
d d d
f x g x f x g x
dx dx dx
 
  
 
       
( ) ( ) ( ) ( )
d d d
f x g x f x g x g x f x
dx dx dx
 
  
 
     
 2
( ) ( ) ( ) ( )
( ) ( )
d d
g x f x f x g x
f x
d dx dx
dx g x g x

 

 
 

3. u u
d du
e e
dx dx
   ln
u u
d du
a a a
dx dx

1
ln
d du
u
dx u dx


4. 1)f(x)=x0.5+0.5x+0.5
𝑓` 𝑥 = 0.5𝑥0.5−1
= 𝟎. 𝟓𝒙−𝟎.𝟓 + 𝟎. 𝟓
=
𝟏
𝟐𝒙𝟎.𝟓
+ 𝟎. 𝟓
=
𝟏
𝟐 𝒙
+ 𝟎. 𝟓
       
( ) ( )
d d d
f x g x f x g x
dx dx dx
 
  
 
   
1
is a real number
n n
d
x nx n
dx


+0.5∗1∗x(1−1)
+𝟎
   
0 is a constant
d
c c
dx


5. 2)f(x)=
𝒄𝒐𝒔𝟐𝒙
𝟐𝒙
𝒇` 𝒙 =
൧
𝟐𝒙 ሺ−𝒔𝒊𝒏𝟐𝒙) ∗ 𝟐 − [ሺ𝒄𝒐𝒔𝟐𝒙) ∗ 𝟐
𝟐𝒙 𝟐
𝒇` 𝒙 =
−𝟒𝒙𝒔𝒊𝒏𝟐𝒙 − 𝟐𝒄𝒐𝒔𝟐𝒙
𝟒𝒙𝟐
𝒇` 𝒙 =
)
𝟐ሺ−𝟐𝒙𝒔𝒊𝒏𝟐𝒙 − 𝒄𝒐𝒔𝟐𝒙
𝟒𝒙𝟐
𝒇` 𝒙 =
𝟐𝒙𝒔𝒊𝒏𝟐𝒙 − 𝒄𝒐𝒔𝟐𝒙
𝟐𝒙𝟐
     
 2
( ) ( ) ( ) ( )
( ) ( )
d d
g x f x f x g x
f x
d dx dx
dx g x g x

 

 
 

6. 3)f(x)=ሺ𝒆𝟐𝒙
+ 𝟓)ሺ𝒍𝒏𝟐𝒙 + 𝟑)
1
ln
d du
u
dx u dx

u u
d du
e e
dx dx

+
𝟏
𝟐𝒙
∗ 𝟐 + 𝟎 𝒆𝟐𝒙
+ 𝟓
൯
f`(x)=ሺ𝒆𝟐𝒙 ∗ 𝟐 + 𝟎 𝒍𝒏𝟐𝒙 + 𝟑
+ 𝒆𝟐𝒙 + 𝟓 ∗
𝟏
𝒙
൯
= 𝒍𝒏𝟐𝒙 + 𝟑 ∗ ሺ𝟐𝒆𝟐𝒙
= 𝟐𝒆𝟐𝒙
𝒍𝒏𝟐𝒙 + 𝟔 𝒆𝟐𝒙 ቇ
+ሺ
𝒆𝟐𝒙
𝒙
+
𝟓
𝒙
       
( ) ( ) ( ) ( )
d d d
f x g x f x g x g x f x
dx dx dx
 
  
 
‫مشتقة‬
‫دالتين‬
‫مضروبة‬
=
(
‫مشتقة‬
‫األول‬
*
‫الثاني‬
(+)
‫مشتقة‬
‫الثاني‬
*
‫األول‬
)

7. 4)f(x)=cos3x.sec2x+sinx2.cot3x
       
( ) ( ) ( ) ( )
d d d
f x g x f x g x g x f x
dx dx dx
 
  
 
f`(x)={[-sin3x*(3)] *sec2x +[sec2x*tan2x*(2)]
* cos3x}
+{[cosx2*(2x)]*cot3x +[- csc23 x*(3)] *sinx2 }
={-3sin3x*sec2x+2sec2x*tan2xcos3x}
+{2xcosx2*cot3x-3csc3x*sinx2}

8. Higher derivatives
Higher derivatives: If a function y = f(x) possesses a derivative at every point of some interval,
we may form the function f '(x) and talk about its derivate, if it has one. The procedure is
formally identical to that used before, that is

9. Example: Find all derivatives of the following
function:

10. Example: Find the third derivative of
the following function:

11. The Chain Rule:

12. Example: Use the chain rule to express
dy / dx in terms of x and y

14. Homework

15. Homework

16. Homework
Use the chain rule to express dy / dx in terms of x and y