Power rule ($d(x^p)/dx=p{x}^{p-1}$)

For a real exponent $p$ and $x>0$, write $x^p=\exp (p\ln x)$. The chain rule and the derivative of the natural logarithm give $d(x^p)/dx=p{x}^{p-1}$. For a nonnegative integer exponent this agrees with polynomial differentiation and extends to all real $x$. Reversing differentiation gives $\int x^{p}dx=x^{p+1}/(p+1)+C$ for $p\ne -1$ on a suitable real branch. The exceptional exponent $p=-1$ gives $\ln |x|+C$ on either half-line.