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Derivative of a complex function
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[2. Mathematical elements.]Derivative of a complex function.The derivative of a function is the rate of change of the value of the function with respect to its argument. There is no substantial difference between the derivative of a real function and the derivative of a complex function, so it is possible to use the usual rules applicable to the derivative of a real function. There are some notations to denote the derivative of a function φ(x): the most common are φ'(x) , dφ(x) / dx , Dφ(x). Here is a small table with the derivative of some functions.
Let φ (x) and Ψ (x) be two complex functions of a real variable. The derivatives of their addition, multiplication and division are:
D (φ (x) + Ψ (x)) =
φ'(x) + Ψ'(x) The derivative φ'(x) of a function φ (x) is a function too; thus there exists also the derivative of φ'(x), which is called the second derivative of φ (x) and is usually denoted by one of the following expressions: φ"(x) , d²φ (x) / dx².
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