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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 '13 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. $\endgroup$ – Daniel Montealegre Sep 27 '13 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... $\endgroup$ – Daniel McLaury Sep 27 '13 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 '13 at 2:28
  • $\begingroup$ Almost surely it was Cauchy. $\endgroup$ – Daniel McLaury Sep 27 '13 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. $\endgroup$ – Arjang Sep 27 '13 at 2:54
  • $\begingroup$ @Arjang: these remarks about Cauchy are not really accurate. See e.g. my answer below. $\endgroup$ – Mikhail Katz Oct 2 '13 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 '13 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? $\endgroup$ – Mikhail Katz Oct 3 '13 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 '13 at 15:51
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Since there's a general unease about infinitesimal methods, which was probably how the power rule was first proven, here's a proof I recently found in an old textbook. If $f(x) = ax^n$, where $a$ is a constant, then: $$f'(x) = \frac{a(x_1^n - x_0^n)}{x_1 - x_0}$$ $$ = a(x_1^{n-1} + x_0x_1^{n-2} + x_0^2x_1^{n-3} + ... + x_0^{n-2}x_1 + x_0^{n-1})$$ Letting $x_1 → x_0$ in value we obtain: $$f'(x) ⇒ nax^{n-1}$$ This approach is usually credited to Caratheodory, but according to the Brian M Scott answer here Cauchy got there first.

EDIT But, the underlying logic was strongly implied by Leibniz:

For instead of the infinite or the infinitely small, one takes quantities as large, or as small, as necessary in order that the error be smaller than the given error, so that one differs from Archimedes' style only in the expressions, which are more direct in our method and conform more to the art of invention.
The Calculus in the Eighteenth Century, p56, HJM Bos.

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