# What is the derivative of sqrt(1/x^3)?

$- \frac{3}{2} {x}^{- \frac{5}{2}}$
Note that $\sqrt{\frac{1}{x} ^ 3}$ is equivalent to $\sqrt{{x}^{- 3}}$ (because $\frac{1}{a}$ is equal to ${a}^{- 1}$). Using the property $\sqrt[a]{{x}^{b}} = {x}^{\frac{b}{a}}$, $\sqrt[2]{{x}^{- 3}} = {x}^{- \frac{3}{2}}$. Our problem now is simply finding the derivative of ${x}^{- \frac{3}{2}}$, which is done easily using the power rule:
$\frac{d}{\mathrm{dx}} {x}^{- \frac{3}{2}} = - \frac{3}{2} \cdot {x}^{- \frac{3}{2} - 1} = - \frac{3}{2} {x}^{- \frac{5}{2}}$