generic regularity of affine varieties

Suppose that $V\subset {\mathbb C}^n$ is an affine subvariety of codimension $p$. How does one prove that $V$ is regular (i.e., is a smooth manifold) at its generic points?

In view of the Jacobian test for regularity (which is just the implicit function theorem in this case), it suffices to show that there exist a point $x\in V$ and polynomials $f_1,…,f_p$ in the defining ideal $I$ of $V$ so that the derivatives $df_1,…, df_p$ are linearly independent at $x$. However, I do not see why such point and polynomials would exist.

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