Derivative of an imaginary number

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Discussion Overview

The discussion revolves around the differentiation of expressions involving imaginary numbers, specifically focusing on the derivative of the term d(ix)/dx and the conditions under which certain functions may have derivatives. The scope includes differential calculus and the treatment of complex numbers as constants or functions.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested, Mathematical reasoning

Main Points Raised

  • One participant inquires about the derivative of an imaginary number, specifically d(ix)/dx.
  • Another participant asserts that for differential calculus, i can be treated as a constant, leading to the conclusion that d(ix)/dx=i.
  • A different viewpoint emphasizes that derivatives are taken of functions rather than numbers, suggesting that complex numbers can be treated as constant functions, thus supporting the earlier claim that d(ix)/dx=i.
  • A participant raises a question about proving the differentiability of the function 1/[z*sin(z)*g(z)] from first principles, introducing the variable z as a complex number.
  • Another participant responds that the differentiability of the function cannot be determined without additional information about g and notes that the function is not differentiable at points where it is undefined, such as multiples of π.
  • A follow-up post suggests a modification to the function under consideration, changing it to 1/[z*sin(z)*cos(z)].

Areas of Agreement / Disagreement

Participants generally agree on the treatment of i as a constant in the context of differentiation, but there is disagreement regarding the differentiability of the proposed functions, particularly due to the undefined nature of certain points.

Contextual Notes

The discussion highlights limitations regarding the lack of information about the function g and the specific conditions under which the proposed functions are defined or undefined.

vbj194
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I was just wondering if anyone knows the rule when taking the derivative of an imaginary number(i). For example: d(ix)/dx=?

Thanks:)
 
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For the purposes of differential calculus, i is simply another constant.
Therefore d(ix)/dx=idx/dx=i
 
You don't take the derivative of "numbers" in general. You take the derivative of functions. Of course you can treat any number, including complex numbers, as a "constant function". As "mathman" said (and he ought to know!) d(ix)/dx= i just as d(ax)/dx= a for any number a.

If you allow the variable, x, to be a complex number, then it becomes more interesting!
 
how can i proof if this function has a derivative?

1/[ z*sin(z)*g(z)] from first principle?

z= x + jy.
 
You don't- not with information on g. And, whatever g is, that function is certainly NOT differentiable where it is not defined: any multiple of [itex]\pi[/itex].
 
suppose to be

1/[ z*sin(z)*cos (z)]
 

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