Who discovered the general rule for differentiating polynomials, in particular that the derivative of $x^n$ is $n x^{n-1}$, and when? I appreciate the answer may not be a clear-cut individual and year, perhaps more a succession of deeper geometric and notational insights. The key point I'm looking for, though, is a dawning awareness that $nx^{n-1}$ represents a *general* rule for the gradient of tangents to polynomials of arbitrary degree, not just *specific* results for the parabola and cubic.

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.

This is generally known as the "[power rule](https://en.wikipedia.org/wiki/Power_rule)", though that might refer to both the integral or derivative of a polynomial. Wikipedia suggests that the power rule for the derivative was discovered by Newton and Leibniz, though that would surprise me. Fermat had clearly explored similar territory — [this student project](http://www.math.wpi.edu/IQP/BVCalcHist/calc2.html) gives some details, but sadly lacks citations, particularly for its key claim that Fermat extended his method to show that the tangent to $y=x^n$ had gradient $n x^{n-1}$. [Isaac Barrow](https://en.wikipedia.org/wiki/Isaac_Barrow) also investigated tangents extensively, using infinitesimal triangles that presaged subsequent developments more closely than Fermat's method, and  [Johannes Hudde](https://en.wikipedia.org/wiki/Johannes_Hudde) worked in this area too, though I know less about his work. His work on polynomials, [particularly that a double-root of a polynomial is also a root of what we would call its derivative, and that a maximum or minimum value occurs at a root of the derivative](https://en.wikipedia.org/wiki/Hudde%27s_rules), comes startlingly close to what I am looking for.

I have sometimes seen the power rule for differentiation called "Wallis'[s] rule" or "Wallis'[s] law", which also suggests an earlier date. However, I have also occasionally seen that phrase attached to [Cavalieri's quadrature formula](https://en.wikipedia.org/wiki/Cavalieri%27s_quadrature_formula) instead — 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*. Effectively Wallis had filled in many of the details of the power rule for integration, [though the exceptional case $n=-1$ was dealt with by others](https://hsm.stackexchange.com/q/2463/3191). Of course, once we are armed with the [Fundamental Theorem of Calculus](https://en.wikipedia.org/wiki/Fundamental_theorem_of_calculus) then the power rules for integration and differentiation are singing the same song, but before this development, the fact somebody could (in modern parlance) integrate $x^n$ does not mean they could differentiate it (or as they would likely see it, find its tangent). Did Wallis know *both* forms of the power rule?

As for "Cavalieri's" formula for integration, I wouldn't be surprised if the answer to my question about differentiation is somewhat different for the separate cases of $n$ being a positive integer, negative integer or rational number.