# How do you find the derivative of sqrt(x^2-1)?

Mar 13, 2016

Use the power rule and chain rule to get $\frac{x}{\sqrt{{x}^{2} - 1}}$.

#### Explanation:

Recognize that $\sqrt{{x}^{2} - 1}$ can be written as ${\left({x}^{2} - 1\right)}^{\frac{1}{2}}$. Now that we have it in this form, we can use the power rule to get:
$\frac{d}{\mathrm{dx}} = \frac{1}{2} {\left({x}^{2} - 1\right)}^{- \frac{1}{2}}$

We also need to use the chain rule; the derivative of the "inside" function (${x}^{2}$) is $2 x$, so we multiply $\frac{1}{2} {\left({x}^{2} - 1\right)}^{- \frac{1}{2}}$ by $2 x$:
$\frac{d}{\mathrm{dx}} = 2 x \cdot \frac{1}{2} {\left({x}^{2} - 1\right)}^{- \frac{1}{2}}$

Canceling the $2$s and using the properties of exponents yields:
$\frac{d}{\mathrm{dx}} = \frac{x}{{\left({x}^{2} - 1\right)}^{\frac{1}{2}}}$

Since the problem was given to us in radical form, convert back to get the final answer:
$\frac{d}{\mathrm{dx}} = \frac{x}{\sqrt{{x}^{2} - 1}}$