# What is the derivative of this function (cos x) / (1-sinx)?

Oct 22, 2017

$\sec x \left(\sec x + \tan x\right) .$

#### Explanation:

Let, $y = \cos \frac{x}{1 - \sin x} .$

$\therefore y = \cos \frac{x}{1 - \sin x} \times \frac{1 + \sin x}{1 + \sin x} ,$

$= \frac{\cos x \left(1 + \sin x\right)}{1 - {\sin}^{2} x} = \frac{\cos x \left(1 + \sin x\right)}{\cos} ^ 2 x .$

$\therefore y = \frac{1 + \sin x}{\cos} x = \frac{1}{\cos} x + \sin \frac{x}{\cos} x , i . e . ,$

$y = \sec x + \tan x .$

$\Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}} = \sec x \tan x + {\sec}^{2} x = \sec x \left(\tan x + \sec x\right) .$