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?
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.
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.