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.
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.