= Solution
For a plane <affine curve> given by a reduced equation $f(x,y)=0$, a point is singular when \b[$f=f_x=f_y=0$] there. For a <projective curve> given by a homogeneous equation $F(x,y,z)=0$, a point is singular when \b[$F=F_x=F_y=F_z=0$] on a nonzero representative. These conditions are invariant under projective rescaling. In characteristic zero, Euler's identity $dF=xF_x+yF_y+zF_z$ makes the equation $F=0$ follow from vanishing of the partials when $d\ne0$ in the field. Over an arbitrary field one should retain the equation itself, particularly if the characteristic divides the degree.
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