Let y = cos(x+y)

Differentitale w.r.t. x, we get

      dy/dx = -sin(x+y)*(1 + dy/dx)

=> dy/dx = -sin(x+y) - sin(x+y)(dy/dx)

=> dy/dx + sin(x+y)(dy/dx) = -sin(x+y)

=> (dy/dx)*(1 + sin(x+y)) = -sin(x+y)

=> dy/dx = -sin(x+y)/{1 + sin(x+y)}

So, differentiation of cos(x+y) w.r.t x is -sin(x+y)/{1 + sin(x+y)}