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", 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 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 also investigated tangents extensively, using infinitesimal triangles that presaged subsequent developments more closely than Fermat's method, and 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, 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 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. Of course, once we are armed with the 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.

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    $\begingroup$ For a similar question on Math Stack Exchange, see Origin of the Power Rule Proof: Who first proved the power rule?. $\endgroup$ – HDE 226868 Nov 3 '15 at 23:33
  • $\begingroup$ Nice catch! I did misread that. Thank you. I'll remove the inaccurate part of my comment. $\endgroup$ – HDE 226868 Nov 3 '15 at 23:49
  • $\begingroup$ @HDE Thanks. I have filled in a few more details. I fear the current state of my question actually contains more information than the related Maths SE answers... though it's somewhat distinct in that I care less about the "proof" aspect, and more about the formulation and generalisation. $\endgroup$ – Silverfish Nov 4 '15 at 0:27
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    $\begingroup$ It would help to focus the question on something more specific, Wikipedia answers all the questions that I can see so far en.wikipedia.org/wiki/Power_rule#History Cavalieri did it for "integrals", Fermat, Torricelli, Roberval and Wallis independently for "derivatives". The -1 power was done by Saint-Vincent and de Sarasa. Of course technically it was all geometric and only reinterpreted as the power rule in hindsight. Explicitly, Newton and Leibniz independently derived the symbolic power rule. $\endgroup$ – Conifold Nov 4 '15 at 1:04
  • $\begingroup$ @Conifold If you read that closely, you'll see that Fermat, Torricelli, Roberval, Wallis, and Pascal are only mentioned in the context of integration (HDE also misread it above, first time round!), and only Newton and Fermat are listed in the context of differentiation - in which sense the Wikipedia article seems to be missing earlier developments. But I take your point about focus. $\endgroup$ – Silverfish Nov 4 '15 at 1:11

Cavalieri was presumably the first to state the "power rule" for areas under parabolas with positive integer exponents, but he only derived it up to $n=4$, beyond that his methods became intractable. "Presumably", because the timeline is spotty due to absence of publications, and Fermat might have done it earlier. In any case Fermat was likely the first to generalize it to rational exponents somewhere between 1635 and 1643. Toricelli, Roberval and Wallis arrived at the same result independently by mid 1640-s. There were no integrals at the time of course, so the rule was expressed and derived in geometric terms, although some derivations, especially Fermat's, anticipated the ideas of Riemann integral.

As for derivatives the name "power rule" is somewhat more tenuous before Newton, who derived it in the familiar form in De Analysi (1669) using fluxions. Since antiquity the tradition was to characterize tangents in terms of so-called subtangents rather than slopes, which in modern terms are segments connecting the abscissa to the $x$-intercept of the tangent. Toricelli considered parabolas with positive integer exponents, and "hyperbolas" $x^my^n=k$ derived their subtangents in terms of $n$ and $m$, which amounts to the power rule for slopes if we help ourselves to the modern notation. But the most interesting part is how he did it. It was not through Fermat's infinitesimals or Descartes-Hudde algebraic calculus, but through kinematic description of the curves as compositions of motions using the parallelogram of velocities that goes back to Aristotle and Archimedes. Archimedes possibly found the tangent to his spiral using such kinematic description, and Toricelli was of course familiar with such description of the ordinary parabola due to his teacher Galileo.

Through Barrow this approach influenced Newton, and became the foundation of his calculus of fluxions. In later works Newton defined quantities as traced by motions, which allowed him to define "first and last ratios" kinematically, without infinitesimals. So Toricelli was Newton's precursor in the same way that Fermat was Leibniz's.

The emergence of the general power rule is discussed in detail in Boyer's History of Calculus, pp.120-172, with references to the original sources.


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