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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