= 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 are
$$
\boxed{\left(\frac{\partial x}{\partial y}\right)_z=\frac{2y(y^2-a)}x,\qquad
\left(\frac{\partial y}{\partial z}\right)_x=\frac{z}{2y(y^2-a)},\qquad
\left(\frac{\partial z}{\partial x}\right)_y=-\frac{x}z.}
$$
Their product is $-1$ wherever $xyz(y^2-a)\ne0$. The excluded factors identify exactly where these quotient formulas cannot all represent finite <partial derivatives>.
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