Past exam of the mathematics course of the University of Cambridge / 2011 / ia / Paper 2 / 6A / b / i / Solution

The partial derivatives of $f$ are $f_x=2x$, $f_y=4y(a-y^2)$ and $f_z=2z$. Therefore the three implicit differentiation formulas areTheir product is $-1$ wherever $xyz(y^2-a)\ne 0$. The excluded factors identify exactly where these quotient formulas cannot all represent finite partial derivatives.