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add wee bit of detail
Silverfish
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Who discovered that the derivative of $x^n$ is $n x ^{n-1}$?

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 — 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.

Silverfish
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