Differentiation rule for power functions

Statement

We have the following differentiation rule:

$\frac{d}{dx}(x^r)=r{x}^{r-1}$

where $r$ is a constant. Some notes on the validity:

Case on $r$	Values of $x$ for which this makes sense
$r=0$	all nonzero $x$. Also makes sense at $x=0$ if we interpret the right side as 0.
$r$ a rational number with odd denominator and greater than or equal to 1	All $x$
$r$ a real number greater than 1 that is not rational with odd denominator	All $x>0$. One-sided derivative makes sense at 0.
$r$ a rational number with odd denominator and between 0 and 1	All $x\ne 0$. At 0, we have a vertical tangent or vertical cusp depending on the numerator of the rational function.
$r$ a real number between 0 and 1 that is not rational with odd denominator	All $x>0$. One-sided vertical tangent at 0.
$r$ a rational number with odd denominator and less than 0	All $x\ne 0$. At 0, we have a vertical asymptote
$r$ a real number less than 0 that is not rational with odd denominator	All $x>0$. One-sided vertical asymptote at 0.