Question

Find the derivative of

sin+cos / sin−cos .​

Answer 1

Step-by-step explanation:

We have,

$y = \frac{ \sin(x) + \cos(x) }{ \sin(x) - \cos(x) }$

Differentiating both sides w.r.t x,

$\frac{dy}{dx} = \frac{ ( \sin(x) - \cos(x) )\frac{d}{dx}( \sin(x) + \cos(x) ) - ( \sin(x) + \cos(x)) \frac{d}{dx} ( \sin(x) - \cos(x) ) }{( \sin(x) - \cos(x))^{2} }$

$\implies \frac{dy}{dx} = \frac{ - ( \sin(x) - \cos(x) )^{2} + ( \sin(x) + \cos(x) )^{2} }{( \sin(x) - \cos(x) )^{2} }$

$\implies \frac{dy}{dx} = \frac{4 \sin(x) \cos(x) }{1 - 2 \sin(x) \cos(x) }$

$\implies \frac{dy}{dx} = \frac{2 \sin(2x) }{1 - \sin(2x) }$