ag.algebraic geometry - Nonsingular point on a hypersurface

Say we are given a hypersurface defined in variables $textbf{x} = (x_1, dots x_n) in mathbb{R}^n$ given by $$P(textbf{x})=0$$ for some homogeneous polynomial P, defined over $mathbb{R}$. I assume this is nondegenerate.

We are given a nonsingular point on this hypersurface $textbf{x}_0$, say.

Question: Is it possible to find a point on the hypersurface arbitrarily close (Euclidean distance) to $textbf{x}_0$ which is nonsingular with respect to a given subset of the variables? I.e. at least one of the partial derivatives from this subset is nonvanishing?

eg can we find an arbitrarily close point $textbf{x}_1$ with $$frac{partial}{partial x_n} P(textbf{x}) Big|_{textbf{x} = textbf{x}_1} neq 0 ?$$ Further could we even find a point arbitrarily close such that NONE of the partial derivatives vanish?

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