If x+y=sinx+y then dydx=

# If $x+y=\mathrm{sin}\left(x+y\right)$ then $\frac{dy}{dx}=$

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### Solution:

Given $x+y=\mathrm{sin}\left(x+y\right)$

Differentiate both sides

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

It implies $1+\frac{dy}{dx}=0⇒\frac{dy}{dx}=-1$

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