Differentiation rule for power functions

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Statement

We have the following differentiation rule:

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

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

Case on r Values of real x for which this makes sense
r = 0 all nonzero x. Also makes sense at x = 0 if we interpret the left side as 1 (constant equal to the list at 0) and 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.