The point P = (-2,1,1) satisfies z^3 + x y^2 z + 1 = 0

Can we implicitly define z in terms of x and y using this equation in a region about P?

Find dz/dx, dz/dy, d2z/dx2 if it makes sense to do so. (note: these are partial derivatives)

I am not sure how to know if we can implicitly define, but is it correct to find partial differentials as follows?

z^3 + x y^2 z + 1 = 0

3z^2 dz/dx + y^2 (x dz/dx + z) = 0

3z^2 dz/dx + xy^2 dz/dx + zy^2 = 0-------------(1)

dz/dx = -zy^2/(3z^2 + xy^2)

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z^3 + x y^2 z + 1 = 0

3z^2 dz/dy + x(2yz + y^2 dz/dy) = 0

dz/dy = -2xyz/(3z^2 + xy^2)

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I can get d2z/dx2 by differentiating (1) and replacing dz/dx by its expression.

Is this method correct? Please let me know how to decide if we can implicitly define z in terms of x and y in a region around P.