# How do you find the derivative of Inverse trig function y = arc csc (x/2)?

Aug 16, 2015

$\frac{2}{{x}^{2} \sin x}$

#### Explanation:

$y = a r c \csc \left(\frac{x}{2}\right)$
$\csc y = \frac{x}{2}$
$\frac{1}{\cos y} = \frac{x}{2}$
$\cos y = \frac{2}{x}$

Diff wrt $x$ on both sides:
$- \sin y \frac{\mathrm{dy}}{\mathrm{dx}} = - \frac{2}{x} ^ 2$
$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{2}{{x}^{2} \sin y}$
$\frac{d}{\mathrm{dx}} a r c \csc \left(\frac{x}{2}\right) = \frac{2}{{x}^{2} \sin y}$