Calculating curvature of a contour

I have a equation of a scalar field in the form


I want to find the curvature of the contour of the curve at fc=f(0.5,0.5).

So I need to calculate the derivative dy/dx and d/dx(dy/dx)

I can solve the equation f(x,y)=fc and get the derivative of f(x,y) w.r.t x

On paper we do,

d/dx (f(x,y))=d/dx(fc) 2*x+2*y*(dy/dx)+y+x(dy/dx)=0 (dy/dx)=-(2*x+y)/(x+2*y) 

and further d/dx(dy/dx) for a curvature approximate

how can I rearrange the equation such that I can get the value of dy/dx on mathematica