# Differentiate Implicitly.sin(x+y)=2x-2y

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### 1 Answer

You need to use implicit differentiation, hence, you need to differentiate the given function with respect to x, as you do when you use the chain rule, such that:

`(d sin(x + y))/(dx) = (d(2x - 2y))/(dx)`

`cos (x + y)*(d(x + y))/(dx) = 2 - 2(dy)/(dx)`

`cos (x + y)*(1 + (dy)/(dx)) = 2 - 2(dy)/(dx)`

You need to open the brackets to the left side, such that:

`cos (x + y) + (dy)/(dx)*cos (x + y) = 2 - 2(dy)/(dx)`

You need to move the terms that contain `(dy)/(dx)` to the left side, such that:

`(dy)/(dx)*cos (x + y) + 2(dy)/(dx) = 2 - cos (x + y)`

You need to factor out `(dy)/(dx)` such that:

`(dy)/(dx)(cos(x+y) + 2) = 2 - cos (x + y)`

`(dy)/(dx)= (2 - cos (x + y))/(2+ cos (x + y))`

**Hence, evaluating (dy)/(dx) using implicit differentiation yields `(dy)/(dx)= (2 - cos (x + y))/(2 + cos (x + y)).` **