= Solution
The notation $(\partial f/\partial x)_y$ means the <derivative> with respect to $x$ while $y$ is held fixed:
$$
\left(\frac{\partial f}{\partial x}\right)_y
=\lim_{h\to0}\frac{f(x+h,y)-f(x,y)}h.
$$
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.
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