# Derivative of an imaginary number

1. Jan 20, 2004

### vbj194

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

Thanks:)

2. Jan 20, 2004

### mathman

For the purposes of differential calculus, i is simply another constant.
Therefore d(ix)/dx=idx/dx=i

3. Jan 23, 2004

### HallsofIvy

Staff Emeritus
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!

4. Sep 2, 2009

### NJunJie

how can i proof if this function has a derivative?

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

z= x + jy.

5. Sep 2, 2009

### HallsofIvy

Staff Emeritus
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 $\pi$.

6. Sep 2, 2009

### NJunJie

suppose to be

1/[ z*sin(z)*cos (z)]