Ministry of Higher Education
and Scientifics research
University of Babylon
Complex Functions
College of Education for Pure Sciences
Physics Department
Three Stage
2.5 Differentiation:Def:- Let a complex function
derivative of function
at
If
EX 1:- If
is defined in region of the
is defined as follows:
, then we can write above equation as follows:-
, then prove that
SOL:-
Ex 2:- Prove that doesn't derivate of
where
Sol:-
We find two limits that is we can't find this limits
1
.
plan the
Ministry of Higher Education
and Scientifics research
University of Babylon
Complex Functions
Thus the derivative of
College of Education for Pure Sciences
Physics Department
Three Stage
isn't exist.
Theorem: - If
is differentiable function then
function . The inverse of this theorem is false.
Def:-Let
, then
where
If
,
is continuous in region . Then
where
as
. And If we write
differential of
. We can write it as
That is
is continuous
then
.
.
NOTES:1) The symbol
2) If
is differential operator.
then
The first derivative of
is
The second derivative of
The
derivative of
is
is
2.6 Rules for Differentiation:Let
are differentiable function at
1)
2
, then
is called
Ministry of Higher Education
and Scientifics research
University of Babylon
Complex Functions
2)
College of Education for Pure Sciences
Physics Department
Three Stage
Where is any constant.
3)
4)
If
5) If
and if
6) If
we get
7) If
and
where is a parameter, then
8) We have also
a.
b.
3