 Projective gradient criterion in characteristic dividing the degree

For a reduced projective hypersurface $X=V(f)$ over an algebraically closed field,The Euler homogeneous function theorem makes the intersection redundant only when the characteristic does not divide the degree. In characteristic $p$, the irreducible polynomial $x_0^p+x_1^{p-1}{x}_2$ has zero gradient at $[1:0:0]$, which is not on $X$.