Past exam of the mathematics course of the University of Cambridge / 2005 / ib / Paper 3 / 3B / i / Solution

Put $r=(x^2+y^2{)}^{1/2}$. Since $x^2{y}^2\le (x^2+y^2{)}^2/4$, the nonzero-point expression satisfiesThus the defining remainder condition for differentiability holds with the zero linear map:The estimate controls every approach direction at once, which is stronger than merely finding the two zero partial derivatives.