= Power rule
{title2=$d(x^p)/dx=px^{p-1}$}
{wiki}
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 $d(x^p)/dx=px^{p-1}$. 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.
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