# Origin of the Power Rule Proof: Who first proved the power rule?

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?

• My money is on either Newton or Leibniz. I don't there is a way to know for sure. Sep 27, 2013 at 1:49
• 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. Sep 27, 2013 at 1:49
• 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... Sep 27, 2013 at 1:59
• 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 Sep 27, 2013 at 2:28
• Almost surely it was Cauchy. Sep 27, 2013 at 2:30

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.