Question

Please answer this. Only genius can answer. Challenge!!​

Answer 1

Given :

$x = log(1 + \sin ^{2} (y ) )$

Differentiate w.r.t y both sides .

$\frac{dx}{dy} = \frac{d( log(1 + { \sin} ^{2}y ) }{dy} \times \frac{d(1 + { \sin }^{2}y }{dy} \\ = > \frac{dx}{dy} = \frac{1}{1 + { \sin}^{2} y} \times (0 + 2 \sin(y) \cos(y) \\ = > \frac{dx}{dy} = \frac{2 \sin(y ) \cos(y) }{1 + { \sin}^{2}y } \\ = > \frac{dx}{dy} =\frac{ \sin(2y) }{1 + { \sin}^{2}y } \\ = > \frac{dy}{dx} = \frac{ {e}^{x} }{ \sin(2y) }$

Formula used :

$\frac{d}{dx} ( log(x) ) = \frac{1}{x } \\ \frac{d}{dx} sinx = cosx \\ \frac{d}{dx} {x}^{n} = n {x}^{n - 1}$

soln also is in attachment