Implicit differentiation is a special case of the chain rule for derivatives. Generally differentiation problems involve functions i.e. y=f(x) - written explicitly as functions of x.
However, some functions like the one given above, are written implicitly as functions of x and y. Here, what we do is to treat y as y=y(x) and use chain rule. This means differentiating y w.r.t. y, but as we have to derive w.r.t. x, as per chain rule, we multiply it by (dy)/(dx).
Thus as x+y-1=ln(x^2+y^2), differentiating w.r.t. x, we have
1+(dy)/(dx)=1/(x^2+y^2)(2x+2y(dy)/(dx))
or 1-(2x)/(x^2+y^2)=(2y)/(x^2+y^2)(dy)/(dx)-(dy)/(dx)
or 1-(2x)/(x^2+y^2)=[(2y)/(x^2+y^2)-1] (dy)/(dx)
or (x^2+y^2-2x)/(x^2+y^2)=[(2y-x^2-y^2)/(x^2+y^2)] (dy)/(dx)
or (dy)/(dx)=(x^2+y^2-2x)/(2y-x^2-y^2)
