= Solution
For each fixed $(x,y)$, the function $z\mapsto z^3+z+xy-1$ is continuous, strictly increasing because its derivative is $3z^2+1>0$, and tends to opposite infinities at the two ends of the real line. The intermediate value theorem therefore gives exactly one zero.
For $F(x,y,z)=z^3+z+xy-1$, its partial derivative $F_z=3z^2+1$ never vanishes. The <implicit function theorem> gives a local $C^1$ solution near every point of the solution set. Uniqueness makes these local solutions agree, producing a global $g\in C^1(\mathbb R^2)$. Implicit differentiation gives
$$
Dg(x,y)=\frac{-1}{3g(x,y)^2+1}\begin{pmatrix}y&x\end{pmatrix}.
$$
At $(1,1)$ the <root> is $g(1,1)=0$, so
$$
\boxed{Dg(1,1)=\begin{pmatrix}-1&-1\end{pmatrix}.}
$$
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