# What is the derivative of sqrt( x^2-1)?

Dec 27, 2015

$\frac{x}{\sqrt{{x}^{2} - 1}}$

#### Explanation:

This is equivalent to ${\left({x}^{2} - 1\right)}^{\frac{1}{2}}$.

According to the chain rule:

$\frac{d}{\mathrm{dx}} \left({u}^{\frac{1}{2}}\right) = \frac{1}{2} {u}^{- \frac{1}{2}} \cdot \frac{\mathrm{du}}{\mathrm{dx}}$

Thus,

$\frac{d}{\mathrm{dx}} \left({\left({x}^{2} - 1\right)}^{\frac{1}{2}}\right) = \frac{1}{2} {\left({x}^{2} - 1\right)}^{- \frac{1}{2}} \frac{d}{\mathrm{dx}} \left({x}^{2} - 1\right)$

$\implies \frac{1}{2 \sqrt{{x}^{2} - 1}} \cdot 2 x$

$\implies \frac{x}{\sqrt{{x}^{2} - 1}}$