Implicit Derivative help

Question

The problem I'm working on is $\sin( x + y ) = 2x-2y$. If anybody could give a step by step solution I would be very appreciative. I'm trying to find the derivative of y

Answer 1

$$\sin( x + y ) = 2x-2y$$ $$\cos(x+y)(1+y')=2-2y'$$ $$\cos(x+y)+y'\cos(x+y)=2-2y'$$ $$y'\cos(x+y)+2y'=2-\cos(x+y)$$ $$y'(\cos(x+y)+2)=2-\cos(x+y)$$ $$y'=\frac{2-\cos(x+y)}{2+\cos(x+y)}$$

Answer 2

I'm assuming you want to find the derivative of $y$ w.r.t. $x$.

You start with $\sin(x+y)=2x-2y$.

Differentiate both sides w.r.t $x$:

$\cos(x+y)\cdot\frac{d}{dx}(x+y)=2-2\frac{dy}{dx}$.

$\cos(x+y)\cdot(1+\frac{dy}{dx})=2-2\frac{dy}{dx}$.

Factor all the terms with $\frac{dy}{dx}$ together:

$(\cos(x+y)+2)\frac{dy}{dx}=2-\cos(x+y)$

Make $\frac{dy}{dx}$ the subject:

$\frac{dy}{dx}=\frac{2-\cos(x+y)}{\cos(x+y)+2}$