Derivative of $f(x)=x^n$

What is the derivative of $$f(x)=x^n$$ when $n$ is a fixed integer and $x$ is a real variable?

As we all know this derivative is

$$f'(x)=nx^{n-1}$$

But for which values of $n$ and $x$ does this hold true?

Well...

1) this is true for integers $n \gt 0$
2) this is also true for integers $n \lt 0$ if $x \ne 0$

Note that when $x=0$ the symbols $x^0, x^{-1}, x^{-2}, x^{-3}, \dots$ are usually treated as undefined. That is why for 2) we need the additional restriction that $x \ne 0$.