Answer
$$\frac{dy}{dx} = \frac{y\sin(x)+\sin(y)}{\cos(x)-x\cos(y)}$$
Work Step by Step
Step-1: Differentiate the following equation with respect to $x$,
$$x\sin (y)=y\cos (x)$$, we get,
$$\sin(y)+x\cos(y)\frac{dy}{dx}=-y\sin(x)+\frac{dy}{dx}\cos(x)$$
Step-2: Isolate $\frac{dy}{dx}$ terms together,
$$\frac{dy}{dx}\big(x\cos(y)-\cos(x)\big) = -y\sin(x)-\sin(y)$$
$$\implies\frac{dy}{dx} = -\frac{y\sin(x)+\sin(y)}{x\cos(y)-\cos(x)}$$
, or,
$$\frac{dy}{dx} = \frac{y\sin(x)+\sin(y)}{\cos(x)-x\cos(y)}$$
