$\text{ If }x+y=\sin (x+y)\text{ then }\frac{dy}{dx}=$

$\begin{array}{l}x+y=\sin (x+y)\\ 1+\frac{dy}{dx}=\cos (x+y)\frac{d}{dx}(x+y)\\ \Rightarrow 1+\frac{dy}{dx}=\cos (x+y)\left(1+\frac{dy}{dx}\right)\\ \Rightarrow 1+\frac{dy}{dx}-\cos (x+y)\left(1+\frac{dy}{dx}\right)=0\\ \Rightarrow \left(1+\frac{dy}{dx}\right)(1-\cos (x+y\left)\right)=0\\ \Rightarrow 1+\frac{dy}{dx}=0\end{array}$

$\Rightarrow \frac{dy}{dx}=-1$