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Who was the first person to prove the power rule for derivatives? The person could have proved the power rule using limits and the binomial theorem or difference of two nth powers, or the implicit differentiation method. I have no preference as to how the person proved it--as long as the person proved it correctly--I'm just wondering: who did it in the first place? Who first proved the power rule?

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    $\begingroup$ My money is on either Newton or Leibniz. I don't there is a way to know for sure. $\endgroup$
    – tylerc0816
    Sep 27, 2013 at 1:49
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    $\begingroup$ Im guessing Newton? but I have no idea. By the way you can also prove that $d/dx(x^n)=nx^{n-1}$ for $n\in\mathbb{N}$ using the product rule and induction. $\endgroup$ Sep 27, 2013 at 1:49
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    $\begingroup$ Whether you mean "proved rigorously, subject to today's standards" or "discovered, understood, and used the result" changes the answer to this question by 100 years or so... $\endgroup$ Sep 27, 2013 at 1:59
  • $\begingroup$ I know Cavalieri found the quadrature formula, and the power rule is the inverse of the quadrature formula, so technically Cavalieri "discovered, understood, and used the result," and also Fermat. So who "proved rigorously, subject to today's standards?" @DanielMcLaury $\endgroup$
    – BrookeW
    Sep 27, 2013 at 2:28
  • $\begingroup$ Almost surely it was Cauchy. $\endgroup$ Sep 27, 2013 at 2:30

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The power rule was already in Fermat, Hudde, Wallis, and Barrow in the 17th century, a few decades before the invention of the calculus by Newton and Leibniz, and two centuries before Cauchy's work in the 19th century (for those who are curious, here is Cauchy's 1821 definition of a continuous function: $f$ is continuous if a change in $x$ by an infinitesimal $\alpha$ necessarily produces an infinitesimal change $f(x+\alpha)-f(x)$ in $y$). Fermat did it using a technique anticipating the calculus called adequality. More details can be found in this recent study.

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A lot of early calculus results were not rigorously proven at that time. For that it wouldn't be a bad idea to look into the live of Cauchy. You know, that man who also developed that mean epsilon-delta approach for the limit? He formalized at lot pertaining calculus what was taken to be for common acknowledgement before that.

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  • $\begingroup$ +1 for mentioning Cauchy, and yes as matter of fact rigor was not enter in math until Cauchy. $\endgroup$
    – jimjim
    Sep 27, 2013 at 2:54
  • $\begingroup$ @Arjang: these remarks about Cauchy are not really accurate. See e.g. my answer below. $\endgroup$ Oct 2, 2013 at 9:29
  • $\begingroup$ @User. Rigor was there before Cauchy's time as well. But to give you an example. Euler's proof about the sum of the reciprocal squares was a wonderful proof, but would not, and I repeat, would NOT stand the test of rigor after Cauchy's contribution in real and complex analysis. He developed epsilon delta approach and that means a great deal in terms of rigor. I am certainly not implying that his predecessors were mathematical "hobbyists" The power rule was probably also proven rigor enough before Cauchy. But rigor without Cauchy is like a car without petrol $\endgroup$
    – imranfat
    Oct 2, 2013 at 15:55
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    $\begingroup$ While you may be correct about Euler's first proof for the Basel problem, he developed other proofs that are as rigorous as anything in Cauchy. This is dealt with in the Euler section of this text. Would it surprise you to find out that Cauchy never gave an epsilon, delta definition of either the limit concept or the concept of continuity of a function? $\endgroup$ Oct 3, 2013 at 8:24
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    $\begingroup$ If you really want to have a sweet surprise about rigor, obtain a translation of Cauchy's Cours d'Analyse (1821). It was the standard of rigor of its time. $\endgroup$
    – imranfat
    Oct 3, 2013 at 15:51

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