Derivative of a complex function
 Complex functions
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Partial derivatives 
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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.
| Function | Derivative | Notes |
| φ (x) = x n | φ'(x) = n x n-1 | n is a natural number |
| φ (x) = sin(x) | φ'(x) = cos(x) | |
| φ (x) = cos(x) | φ'(x) = - sin(x) | |
| φ (x) = e x | φ'(x) = e x | e is the base of natural logarithms (e=2.718...) |
| φ (x) = e - x | φ'(x) = - e - x | |
| φ (x) = e k x | φ'(x) = k e k x | |
| φ (x) = e i x | φ'(x) = e i (x+π/2) | |
| φ (x) = e - i x | φ'(x) = - e -i (2x+π) / 2 | |
| φ (x) = e k i x | φ'(x) = k e i (kx+π/2) | |
| φ (x) = ln(x) | φ'(x) = 1/x | ln is the natural logarithm |
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)
D (φ (x) · Ψ (x)) =
φ (x) · Ψ'(x) +
φ'(x) · Ψ (x)
D (φ (x) / Ψ (x)) =
φ'(x) / Ψ (x) -
Ψ'(x) · φ (x) / (Ψ (x))²
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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Complex functions
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Partial derivatives
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