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