Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2003 / Paper 2 / 10 / a / Solution

Away from the origin, the chain rule gives $r_x=x/r$ and $r_y=y/r$. Differentiating once more yields the two-dimensional case of the Hessian of the Euclidean norm:These partial derivatives are smooth wherever $r>0$, so Clairaut's theorem gives $\boxed{r_{yx}=r_{xy}=-xy/r^3}$. None of these formulas supplies a derivative at the origin: even the restriction $r(x,0)=|x|$ has no two-sided derivative there.