Past exam of the mathematical methods II course of the University of Oxford / 2016 resit / 12 / b

The notation $(\partial f/\partial x{)}_y$ means the derivative with respect to $x$ while $y$ is held fixed:For a differentiable function, its first-order height change is $df=f_{x}dx+f_{y}dy$. If both coefficients vanish, the tangent plane is horizontal and every directional derivative vanishes: the surface is flat to first order at that point. It need not be constant in a neighbourhood; $f=x^2+y^2$ at the origin is a counterexample. Existence of coordinate partials alone, without differentiability, would not justify this tangent-plane conclusion.