Why is the point at infinity of the affine equation $x^2 + y^2 = 1 - (xy)^2$ singular?

Question

The equation of the plane projective curve is $f(x,y,z) = (xz)^2 + (yz)^2 - z^4 + (xy)^2$. Two points at infinity are $(1:0:0)$ and $(0:1:0)$. The definition says that a point $P$ is singular if $\frac{\partial f}{\partial x}$, $\frac{\partial f}{\partial y}$, $\frac{\partial f}{\partial z}$ all vanish at $P$. But $\frac{\partial f}{\partial x} = 2x(y^2 + z^2)$, which I think exists at $(1:0:0)$, so where did I start going wrong?