#math/calculus For every integer $n$, $ \frac{d}{dx}x^{n}=nx^{n-1}, $ where $x\neq 0$ when $n<0$. Positive powers follow from repeated use of the product rule. If $n=-m<0$, the reciprocal rule gives $ \frac{d}{dx}x^{-m} =\frac{d}{dx}\left(\frac{1}{x^{m}}\right) =-\frac{mx^{m-1}}{x^{2m}} =-mx^{-m-1}. $ The case $n=0$ is the constant rule: $(x^{0})'=1'=0$ on $x\neq 0$. > [!Practice] > Marsden & Weinstein, p. 89, exercises 67–68.