Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2013 / Paper 2 / 18 / a / Solution

Along a differentiable graph satisfying the implicit equation, the multivariable chain rule givesThus, wherever $F_y\ne 0$,The implicit function theorem supplies such a local graph under that nonvanishing condition. At a singular point both numerator and denominator vanish, so this division cannot determine its tangent.