Is this correct?

Let F(x,y,z) = z^3 + xy^2 z + 1

F(x,y,z) is continuous at P.

Partial derivative dF/dz = 3z^2 + xy^2

At P: dF/dz = 3(1)^2 + (-2)1^2 = 1 (not= 0)

Therefore the implicit definition is possible.

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

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

Is this correct? But how to find the d2z/dx2 ?

Note: I have used 'd' here but it is actually 'del' (partial derivative)