= Solution
The <Jacobian criterion> on an affine chart $x_j\ne0$ says that a point of the <projective hypersurface> is singular exactly when all <derivatives> of its dehomogenized equation vanish there. The <Euler homogeneous function theorem> identity $\sum_ix_if_{x_i}=df$ then recovers the omitted $j$th <derivative> on $X$, without dividing by $d$. Thus the correct statement in every characteristic is
$$
\boxed{\operatorname{Sing}(X)=X\cap V(f_{x_0},\ldots,f_{x_n}).}
$$
If the printed $Z(I)$ denotes a zero locus inside $X$, this proves it as written. If it means the ambient projective zero locus, the assertion is false when the characteristic divides $d$. When the characteristic does not divide $d$, the <Euler homogeneous function theorem> forces $f=0$ at every common zero of the <derivatives>, and the intersection with $X$ is redundant.
For a genuine counterexample to the unrestricted ambient interpretation, in characteristic $p$ take
$$
f=X_0^p+X_1^{p-1}X_2.
$$
This is an irreducible <homogeneous polynomial>: it is primitive and linear in $X_2$ over $k[X_0,X_1]$, with coprime coefficients. At $[1:0:0]$ all <partial derivatives> vanish, including when $p=2$, but $f=1$. This point is outside $X$. Hence \b[the ambient-locus version needs the displayed intersection or a restriction on the degree].
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