What is $\frac{d}{dx}x^i$?

Question

What is $\frac{d}{dx}x^i$ where $i=\sqrt{-1}$? Does the proof as for real indices apply in the same way here?

Answer 1

The rule does apply, so the derivative is $ix^{i-1}$.

For a simple proof,

$\dfrac{d}{dx}x^i$

$=\dfrac{d}{dx}e^{i\ln x}$

$=e^{i\ln x}\dfrac{d}{dx}i\ln x$

$= e^{i\ln x}\cdot\dfrac{i}{x}$

$ =i\dfrac{e^{i\ln x}}{e^{\ln x}} $

$=ie^{(i-1)\ln x}=ix^{i-1}$

Answer 2

$i$ is just treated as a constant, so the power rule applies.