# How do you find the derivative of  csc^-1 (u)?

##### 1 Answer
Feb 27, 2017

$\frac{\mathrm{dx}}{\mathrm{du}} = - \frac{1}{u \sqrt{{u}^{2} - 1}}$

#### Explanation:

Let $x = {\csc}^{-} 1 u$

$u = \csc x$

$\frac{\mathrm{du}}{\mathrm{dx}} = - \csc x \cot x$

$1 + {\cot}^{2} x = {\csc}^{2} x = {u}^{2}$

$\cot x = \sqrt{{u}^{2} - 1}$

$\frac{\mathrm{du}}{\mathrm{dx}} = - u \sqrt{{u}^{2} - 1}$

$\frac{\mathrm{dx}}{\mathrm{du}} = - \frac{1}{u \sqrt{{u}^{2} - 1}}$