Who discovered the general rule for differentiating polynomials, in particular that the derivative of $x^n$ is $n x^{n-1}$, and when?

I'm not so concerned *linearity*, which allows us to differentiate polynomials with several terms, nor at what point the discovery was *proven* (to whatever degree of rigour), but if someone wants to include such detail in the answer for completeness that would, of course, be welcome.

I have sometimes seen this called "Wallis'[s] rule" or "Wallis'[s] law", but I have also occasionally seen that phrase attached to [Cavalieri's quadrature formula](https://en.wikipedia.org/wiki/Cavalieri%27s_quadrature_formula) — which would be fair enough, since according to Wikipedia, Wallis extended Cavalieri's discovery (in modern notation, that $\int_0^a x^n \mathrm{d}x = \frac{1}{n+1}a^{n+1}$, for $n \in \mathbb{N}$) to rational and negative indices $n$ in his work *Arithmetica Infinitorum*.

Similarly, I wouldn't be surprised if the answer to my question is somewhat different for the separate cases of $n$ being a positive integer, negative integer or rational number.