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

##### 1 Answer
Jan 12, 2016

$\frac{x}{1 - {x}^{2}} ^ \left(\frac{3}{2}\right)$ or $x {\left(1 - {x}^{2}\right)}^{- \frac{3}{2}}$

#### Explanation:

This can be done basically using the chain rule. This is easier to understand if you use a substitution.
let $t = \left(1 - {x}^{2}\right)$
so $\frac{\mathrm{dt}}{\mathrm{dx}} = - 2 x$
Then $f \left(t\right) = \frac{1}{\sqrt{t}} = {t}^{- \frac{1}{2}}$
$\frac{\mathrm{df}}{\mathrm{dt}} = - \frac{1}{2} \cdot {t}^{- \frac{3}{2}}$
$\frac{\mathrm{df}}{\mathrm{dx}} = \frac{\mathrm{df}}{\mathrm{dt}} \cdot \frac{\mathrm{dt}}{\mathrm{dx}}$
$= - \frac{1}{2 {t}^{\frac{3}{2}}} \cdot - 2 x$
$= \frac{x}{1 - {x}^{2}} ^ \left(\frac{3}{2}\right)$ or $x {\left(1 - {x}^{2}\right)}^{- \frac{3}{2}}$