z = y/(x^2 - y^2)
dz/dx = (y/(x^2 - y^2))'_x = y * ((x^2 - y^2)^(-1))'_x =
= y * (-1) * 1/(x^2 - y^2)^2 * (x^2 - y^2)'_x = -2x * y/(x^2 - y^2)^2
dz/dy = (y/(x^2 - y^2))'_y = 1/(x^2 - y^2) + y * ((x^2 - y^2)^(-1))'_y =
= 1/(x^2 - y^2) + y * (-1) * 1/(x^2 - y^2)^2 * (x^2 - y^2)'_y =
= 1/(x^2 - y^2) + 2 * y^2/(x^2 - y^2)^2
Подставим в исходное уравнение:
1/x * dz/dx = 1/x * (-2x * y/(x^2 - y^2)^2) = -2y/(x^2 - y^2)^2
1/y * dz/dy = 1/y * (1/(x^2 - y^2) + 2 * y^2/(x^2 - y^2)^2) =
= 1/(y * (x^2 - y^2)) + 2y/(x^2 - y^2)^2

1/x * dz/dx + 1/y * dz/dy - z/y^2 =
= -2y/(x^2 - y^2)^2 + 1/(y * (x^2 - y^2)) + 2y/(x^2 - y^2)^2 - (y/(x^2 - y^2))/y^2 =
= 1/(y * (x^2 - y^2)) - 1/(y * (x^2 - y^2)) = 0,
что и требовалось доказать.