Find $\frac{dy}{dx}$, if $y + \sin y = \cos x$.

$\frac{-\sin x}{1 + \cos y}$

The correct answer is Option (2) → $\frac{-\sin x}{1 + \cos y}$ ##

We differentiate the relationship directly with respect to $x$, i.e.,

$\frac{dy}{dx} + \frac{d}{dx}(\sin y) = \frac{d}{dx}(\cos x)$

which implies using chain rule

$\frac{dy}{dx} + \cos y \cdot \frac{dy}{dx} = -\sin x$

This gives $\frac{dy}{dx} = -\frac{\sin x}{1 + \cos y}$

Where $y \neq (2n + 1)\pi$