Question

$y=\sqrt{\sin x+\sqrt{\sin x+\sqrt{x}+....\infty }}$
prove that $\frac{dy}{dx}=\frac{\cos x}{2y-1}$

Answer 1

$y=\sqrt{\sin x+y}$

$y^2=\sin x+y$

$2y\frac{dy}{dx}=\cos x+\frac{dy}{dx}$

$(2y-1)\frac{dy}{dx}=\cos x\Rightarrow \frac{dy}{dx}=\frac{\cos x}{2y-1}$