THE DERIVATIVE OF A FUNCTIONS
LEC :
1
1
MOHAMED SABAH AL TAEE \ MOSUL UNIVERSITY \ MATHEMATICS SCIENCE
is the slope and we can define as :
:
The Derivative
0
(
)
( )
( )
lim
x
dy
f x
x
f x
f x
m
dx
x
Where " m " is the slope , if this derivative exists then we say f is " differentiable " .
( )
f x
x
dy
dx
to the function
d
Fin
:
EXAM
Solution :
0
0
0
0
0
0
(
)
( )
( )
lim
lim
lim
lim
lim
(
)
(
)
1
1
1
lim
2
x
x
x
x
x
x
dy
f x
x
f x
f
x
dx
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
to the function
dy
dx
d
Fin
:
EXAM
1)
2
( )
f x
x
Solution :
2
2
2
2
0
0
(
)
(
)
2
(
)
( )
( )
lim
lim
x
x
f x
x
x
x
x
x x
x
f x
x
f x
x
f
x
x
2
2
2x x
x
x
2
0
0
2
lim
lim
x
x
x
x x
x
x
x
[2
]
x
x
x
0
lim 2
2
0
2
x
x
x
x
x
2)
1
( )
f x
x
Solution :
1
(
)
f x
x
x
x
THE DERIVATIVE OF A FUNCTIONS
LEC :
1
2
MOHAMED SABAH AL TAEE \ MOSUL UNIVERSITY \ MATHEMATICS SCIENCE
0
0
2
0
1
1
(
)
( )
lim
lim
1
1
lim
(
)
x
x
x
x
x
x
x x
x
x
x
x
f
x
x
x
x x
x
x
RULES OF DERIVATIVE :
1
2
1)
( )
0
2)
( )
1
3)
( )
4)
( )
( )
( )
5)
( )
( )
( )
( )
6)
( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
7)
( )
( )
8)
( )
( ( ))
n
n
n
y
f x
C
y
y
f x
x
y
y
f x
x
y
nx
y
f x
Cg x
y
Cg x
y
f x
g x
y
f
x
g x
y
f x
g x
y
f x g x
g x f
x
f x
g x f
x
f x g x
y
y
g x
g x
y
f x
C g x
y
C
1
( ( ))
( )
n
g x
g x
:
Proof
0
0
0
1)
( )
(
)
(
)
( )
( )
lim
lim
0
2)
( )
(
)
( )
lim
x
x
x
f x
C
f x
x
C
f x
x
f x
C
C
f
x
x
x
f x
x
f x
x
x
x
x
f
x
x
x
0
lim
1
x
x
x
x
1
2
2
3
3
(
1)
(
1)(
2)
(
)
2!
3!
n
n
n
n
n
n
NOTE
n n
n n
n
x
y
x
nx
y
x
y
x
y
y
THE DERIVATIVE OF A FUNCTIONS
LEC :
1
3
MOHAMED SABAH AL TAEE \ MOSUL UNIVERSITY \ MATHEMATICS SCIENCE
1
2
2
0
3)
( )
1
(
)
(
)
...
2!
( )
lim
n
n
n
n
n
n
n
x
f x
x
n n
f x
x
x
x
x
nx
x
x
x
x
x
f
x
1
2
2
1
...
2!
n
n
n
n
n n
nx
x
x
x
x
x
0
lim
x
x
x
1
1
2
1
1
[
...
]
2!
n
n
n
n n
nx
x
x
x
x
1
1
2
1
0
1
1
1
lim
...
2!
0
0 ... 0
n
n
n
x
n
n
n n
nx
x
x
x
nx
nx
d
Fin
:
EXAM
dy
dx
to the function
6
5
4
3
3
4
2
3
2
2
2
2
2
2
2
5
2
4
2
5
3
3
1)
6
2)
3
3.4
12
3)
3
2
7
10
12
4
7
4)
(
3)(
2)
(
3)(2 )
(
2).1
(
3)
(
2).1 (
3)(2 )
5)
(
2)
(
2)
6)
(3
4 )
5(3
4 ) (6
4)
7)
(3
4 ) (
7)
y
x
y
x
y
x
y
x
x
y
x
x
x
y
x
x
y
x
x
y
x
x
x
x
x
x
x
y
y
x
x
y
x
x
y
x
x
x
y
x
x
x
2
5
3
2
2
3
3
2
4
(3
4 ) .3(
7) .(3
)
(
7) 5(3
4 ) (6
4)
y
x
x
x
x
x
x
x
x
االشتقاق الضمني
IMPLICIT DIFFERENTIATION
حالةٍف ٍىمضلا قاقتشلاا مدختسو
إعطاء
ه اوَرُغتمب ةلادلا
أكثر
رر المسرتقُغتملا قتشو ثُح ,
Independent
) اشرتقااا
ر المعتمدُغتملا اما , احَرص
Dependent
)
ثُح , اُىمض اااقتشا هقتشو
أن
ًعتمرد ع رَ ٌيرلا رُغتملا ىه دمتعملا رُغتملا
: شتق بالىسبة له . مثَُ ٌيلا رُغتملا ىهف قتسملا رُغتملا اما يرخا تارُغتم
(dependent)
& x (independent)
(dependent)
& y (independent)
dy
y
dx
dz
z
dy
THE DERIVATIVE OF A FUNCTIONS
LEC :
1
4
MOHAMED SABAH AL TAEE \ MOSUL UNIVERSITY \ MATHEMATICS SCIENCE
:
EXAM
2
2
2
2
2
2
2
2
2
2
2
2
3
3
9
,
&
:
2
2
0
2
2
( .1
)
9
dy
d y
If
x
y
Find
dx
dx
Solution
x
x
yy
yy
x
y
y
x
x
y
y
y
xy
y
y
y
y
y
y
x
y
y
y
:
EXAM
2
2
3
2
2
2
4
2
4
2
2
2
2
2
3
2
2
3
2
3
3
2
3
3
2
,
8
:
3
4
3
4
3
8
3
2
3
.2
.
3
3
4
4
4
4
4
3 (8
3
)
3
(8
6
)
2
4
16
4
3
(2
)
3
16
8
d y
x
If
y
x
then prove that
y
dx
solution
x
yy
x
y
y
x
xy
x
yx
y
x
x
y
y
y
y
y
y
y
x
y
x
x
y
y
x
y
y
y
x
y
x
y
y
:
EXAM
3
3
3
2
3
2
5
,
:
2
If
y
x
a
prove that
d x
ya
dy
x
a is constant.
THE DERIVATIVE OF A FUNCTIONS
LEC :
1
5
MOHAMED SABAH AL TAEE \ MOSUL UNIVERSITY \ MATHEMATICS SCIENCE
2
2
2
2
2
2
2
4
2
2
2
2
2
2
4
4
4
3
4
3
3
3
3
5
5
5
5
3
3
0
3
3
2
2
.2
2
.2
.2
2
2
2 (
)
2 (
)
2
y
y
x x
y
x x
x
x
y
y
yx
xy
yx
x
y
y
xx
x
x
x
x
x
x
yx
y
y x
y
y
a
ya
x
x
x
x
قاعدة السلسلة
CHAIN RULE
Let y=f(t) and g(t) , the chain rule may be written as :
.
dy
dy
dy dt
dt
dx
dx
dt dx
dt
:
EXAM
3
2
2
6
,
2
4
,
If y
t
x
t
dy
d y
Find
dx
dx
ٍ
Solution :
2
2
2
2
2
.
1
3
,
2
2
1
3
3 .
2
2
(
)
.
1
3
3 .
2
2
dy
dy dt
dx
dt dx
dy
dx
dt
t
dt
dt
dx
dy
t
t
A
dx
d y
d
dy
d
dA
dA dt
A
dx dx
dx
dx
dt dx
dx
t
t
THE DERIVATIVE OF A FUNCTIONS
LEC :
1
6
MOHAMED SABAH AL TAEE \ MOSUL UNIVERSITY \ MATHEMATICS SCIENCE
:
EXAM
2
2
2
2
1 ,
1
,
If y
t
x
t
dy
d y
Find
dx
dx
ٍ
Solution :
2
2
.
1
2
,
2
2
1
2 .
1
2
(
)
.
1
0.
0
2
dy
dy dt
dx
dt dx
dy
dx
dt
t
t
dt
dt
dx
t
dy
t
A
dx
t
d y
d
dy
d
dA
dA dt
A
dx dx
dx
dx
dt dx
dx
t
:
EXAM
2
2
,
sin
,
If y
Cost
x
t
dy
d y
Find
dx
dx
ٍ
Solution :
2
2
2
3
.
1
sin
,
1
sin .
tan
(
)
.
1
.
dy
dy dt
dx
dt dx
dy
dx
dt
t
Cost
dt
dt
dx
Cost
dy
t
t
A
dx
Cost
d y
d
dy
d
dA
dA dt
A
dx dx
dx
dx
dt dx
dx
Sec t
Sec t
Cost
THE DERIVATIVE OF A FUNCTIONS
LEC :
1
7
MOHAMED SABAH AL TAEE \ MOSUL UNIVERSITY \ MATHEMATICS SCIENCE
Partial Derivative :
DEF:
Let f be a function with two variables x, y , then the partial derivative for f respect to x
and its value at any point (x,y) in domain f is :
f
x
is the function f
x
or
0
(
, )
( , )
( , )
( , )
lim
x
x
f x
x y
f x y
f
x y
f x y
x
x
DEF:
Let f be a function with two variables x, y , then the partial derivative for f respect to x
and its value at any point (x,y) in domain f is :
f
y
is the function f
x
or
0
( ,
)
( , )
( , )
( , )
lim
y
y
f x y
y
f x y
f
x y
f x y
y
y
DEF:
Let f be a function with three variables x, y ,z then the partial derivative for f :
0
0
0
(
, , )
( , , )
( , , )
( , , )
lim
( ,
, )
( , , )
( , , )
( , , )
lim
( , ,
)
( , , )
( , , )
( , , )
lim
x
x
y
y
z
z
f x
x y z
f x y z
f
x y z
f x y z
x
x
f x y
y z
f x y z
f
x y z
f x y z
y
y
f x y z
z
f x y z
f
x y z
f x y z
z
z
Note :
2
2
2
2
2
2
xx
xy
yx
yy
xy
yx
f
f
f
f
xy
x
f
f
f
f
yx
y
f
f
and in three variables
3
2
2
2
(
)
.
xyy
f
f
f
etc
x
x y
y
EXAM :
2
3
( , )
5
f x y
x
xy
y
2
2
2
5
0 5
3
5
3
f
x
y
x
f
x
y
x
y
y
THE DERIVATIVE OF A FUNCTIONS
LEC :
1
8
MOHAMED SABAH AL TAEE \ MOSUL UNIVERSITY \ MATHEMATICS SCIENCE
2
2
2
2
2
2
(
)
6
(
)
2
5
f
f
y
y
y
y
f
f
x
x
x
f
f
x y
y x
EXAM :
5
4
4
4
4
2
3
3
2
2
3
3
2
2
2
3
3
( , )
(3
5 )
5(3
5 ) .3 15(3
5 )
5(3
5 ) .5
25(3
5 )
60(3
5 ) .3 180(3
5 )
100(3
5 ) .5
500(3
5 )
60(3
5 ) .5
300(3
5 )
f x y
x
y
f
x
y
x
y
x
f
x
y
x
y
y
f
x
y
x
y
x
f
x
y
x
y
y
f
f
x
y
x
y
x y
y x
EXAM :
3
2
4
3
4
2
2
4
2
3
2
4
2
2
2
2
2
2
2
( , )
5
10
3
5
20
10
6
60
f x y
xy
x yz
f
y
xyz
x
f
y x
x z
y
f
x yz
z
f
yz
x
f
xy
y
f
x yz
z
3
3
3
3
3
2
3
2
2
4
2
3
3
3
2
2
2
0
6
120
3
10
40
(
(
))
(
)
0
f
x
f
x
y
f
x yz
z
f
y
xz
x y
f
xyz
x z
f
f
f
x y z
x
y
z
f
f
z
z y
y
THE DERIVATIVE OF A FUNCTIONS
LEC :
1
9
MOHAMED SABAH AL TAEE \ MOSUL UNIVERSITY \ MATHEMATICS SCIENCE
DEF :
Let u(x,y) , v(x,y) be a functions then we say that u & v satisfy the Cauchy Riemann
Equation (C.R.E) iff
&
u
v
u
v
x
y
y
x
EXAM :
2
2
,
2
2
,
2
2
,
2
&
. .
u
x
y
v
xy
u
v
u
v
x
x
x
y
x
y
u
v
u
v
y
y
y
x
y
x
u
v
satisfy C R E
DEF :
2
2
2
2
0
f
f
x
y
When the function f(x,y) satisfy Laplace equation
then f(x,y) is said to be harmonic .
EXAM :
3
2
2
2
2
2
2
2
2
2
2
2
( , )
3
3
3
6
6
6
6
6
0
f x y
x
xy
f
f
x
y
x
x
x
f
f
xy
x
y
y
f
f
x
x
x
y
f
harmonic
THE DERIVATIVE OF A FUNCTIONS
LEC :
1
11
MOHAMED SABAH AL TAEE \ MOSUL UNIVERSITY \ MATHEMATICS SCIENCE
to
( )
f
x
By using the definition of the derivative find
3
2
5
2
3
,
1
,
1
1
,
,
1
1
x
x
x
x
x
x
2
2
2
3
3
3
1
1
1
3
3
3
2
2
2
2
2
2
2
3
3
2
&
:
7
2
4
1
0
,
.
,
.
4
3
0
d y
d x
Find
to the
dx
dy
x y
y
x y
y
x
y
a
a is cons
x
y
a
a is cons
x
y
3
2
4
2
2
2
3
1
4
2
If y
t
t
x
t
t
d y
then Find
dx
1
2
3
MOHAMED SABAH MAHMOUD AL TAEE
M.SC / MATHEMATICS
E-MAIL : msmt_80@yahoo.com
2013 -2014