If y = 2 + sin x + 2 + sin x + 2 + sin x + ∞ , then the value of...

Given equation can be rewritten as
$y = 2 + \sqrt{sin x + y}$
$⇒ \left(y - 2\right)^{2} = sin x + y$
$⇒ y^{2} - 4y + 4 = sin x + y \ldots \left(i\right)$
Putting $x = 0$ in the equation $\left(i\right)$, we get,
$y^{2} - 4y + 4 = 0 + y ⇒ y^{2} - 5y + 4 = 0$
$⇒ \left(y - 1\right)$\left(y - 4\right) =0$<br />$y = 1 or y = 4$<br />$∵ y > 2 ⇒ y = 4$<br />Now differentiating equation \left(i\right)$ with respect to $x$, we get,
$\dfrac{dy}{dx} = \dfrac{cos x}{2y - 5}<br />Putting$x = 0$,$y = 4$</p><p>\dfrac{dy}{dx} =\dfrac{\cos \left(0\right)}{2\left(4\right)- 5} = \dfrac{1}{3}</p>$