Verified answer

Differentiate both sides with respect to x (holding y constant):

(1/(xy + yz)) * (y + y ∂z/∂x) = 2x + 2z ∂z/∂x.

==> (1/(x+z)) * (1 + ∂z/∂x) = 2x + 2z ∂z/∂x

==> 1 + ∂z/∂x = 2x(x + z) + 2z(x + z) ∂z/∂x

==> ∂z/∂x = [2x(x + z) - 1]/[1 - 2z(x + z)]

................= (2x^2 + 2xz - 1)/(1 - 2xz - 2z^2).

So, the quotient rule yields

∂^2z/∂x^2 = [(4x + 2z + 2x ∂z/∂x)(1 - 2xz - 2z^2) - (2x^2 + 2xz - 1)(-2z - 2x ∂z/∂x - 4z ∂z/∂x)]

/ (1 - 2xz - 2z^2)^2.

Substitute ∂z/∂x = (2x^2 + 2xz - 1)/(1 - 2xz - 2z^2) into the above result and simplify for the final result.

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As for ∂z/∂y, differentiate both sides of the original equation with respect to y instead of x:

(1/(xy+yz)) * (x ∂z/∂y + z + y ∂z/∂y) = 2y + 2z ∂z/∂y.

Solve for ∂z/∂y as before.

(x+y) ∂z/∂y + z = 2y(xy+yz) + 2z(xy+yz) ∂z/∂y

==> ∂z/∂y = [2y(xy+yz) - z]/((x+y) - 2z(xy+yz)).

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I hope this helps!