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differentiate wrtx

under root (1+sinx/1-sinx)

$y = \sqrt{\frac{1 + \sin x}{1 - \sin x}} = \sqrt{\frac{\sin^{2} (\frac{x}{2}) + \cos^{2} (\frac{x}{2}) + 2 \sin (\frac{x}{2}) \cos (\frac{x}{2})}{\sin^{2} (\frac{x}{2}) + \cos^{2} (\frac{x}{2}) - 2 \sin (\frac{x}{2}) \cos (\frac{x}{2})}} = \sqrt{\frac{(\sin \frac{x}{2} + \cos \frac{x}{2})^{2}}{{(\sin \frac{x}{2} - \cos \frac{x}{2})}^{2}}} y = \frac{\sin \frac{x}{2} + \cos \frac{x}{2}}{\sin \frac{x}{2} - \cos \frac{x}{2}} = \frac{1 + \tan \frac{x}{2}}{1 - \tan \frac{x}{2}} = \tan (\frac{π}{4} + \frac{x}{2}) \frac{d y}{d x} = \frac{1}{2} \sec^{2} (\frac{π}{4} + \frac{x}{2}) = \frac{1}{2 \cos^{2} (\frac{π}{4} + \frac{x}{2})} = \frac{1}{2 (\frac{1}{2}) (\cos \frac{x}{2} - \sin \frac{x}{2})^{2}} = \frac{1}{1 - \sin x} = \frac{1}{1 - \sin x} \times \frac{1 + \sin x}{1 + \sin x} = \frac{1 + \sin x}{1 - \sin^{2} x} = \frac{1 + \sin x}{\cos^{2} x} = \sec x (\sec x + \tan x)$

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