Partial Derivative wrt z of z^2 equals x^2 - 2 x y - 1 at (1, -2, -2)

Theorem
Let $z^2 = x^2 - 2 x y - 1$.

Then:
 * $\valueat {\dfrac {\partial z} {\partial x} } {x \mathop = 1, y \mathop = -2, z \mathop = -2} = -\dfrac 3 2$

Proof
First we make sure that $\tuple {1, -2, -2}$ actually satisfies the equation:

and it is seen that this is the case.