 Solution

ID: 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 gives

\frac{d}{d x} F (x, y (x)) = F_{x} + F_{y} \frac{d y}{d x} = 0.

Thus, wherever

F_{y} \neq = 0

\frac{d y}{d x} = - \frac{F _{x}}{F _{y}} .

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.

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