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If I have some function $f = (1 + 7i - x)(7 + 5i - x)(3 + 1i - x)$

How do I find its derivative? I know it is $(13/3 + 11i/3 - x)(3 + 5i - x)$ but I don't know how to find it.

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Exactly like usual differentiation. Multiply out and differentiate (unpleasant) or use the product rule twice. The number $i$ is just a constant. – André Nicolas May 26 '12 at 6:29
But I thought all constants drop to 0 in differentiation? – John Smith May 26 '12 at 6:31
Indeed they do. By the way, the answer you give is not correct. – André Nicolas May 26 '12 at 6:33
The solution seems to be fine (I verified it on Wolfram) -- it seems that what I am after here are the complex roots of the derivative. Complex roots are 13/3+11i/3 and 3+5i – John Smith May 26 '12 at 6:35
The roots may be right. But the coefficient of $x^2$ in the derivative should be $-3$, and the coefficient of $x^2$ in the answer you give is $1$. – André Nicolas May 26 '12 at 6:38
up vote 2 down vote accepted

Expand the function $f$:

$$f(x) = -x^3+(11+13i)x^2+(16-98i)x-138+134i.$$ Then differentiate as usual.

Factoring the derivative gives:

$$f'(x) = (3+5i-x)(3x-13-11i).$$

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