If $z = \tan^{-1}\left(\frac{x}{y}\right)$,then $z_x : z_y = $

The least value of $2x^{2} + y^{2} + 2xy + 2x - 3y + 8$ for real numbers $x$ and $y$ is:

If $u = \tan^{-1}\left(\frac{x^3 + y^3}{x - y}\right)$,then $x\frac{\partial u}{\partial x} + y\frac{\partial u}{\partial y} = $

If $u = \log (x^3 + y^3 + z^3 - 3xyz)$,then $\left( \frac{\partial u}{\partial x} + \frac{\partial u}{\partial y} + \frac{\partial u}{\partial z} \right) (x + y + z) =$

If $z=\log (\tan x+\tan y)$,then $(\sin 2 x) \frac{\partial z}{\partial x}+(\sin 2 y) \frac{\partial z}{\partial y}$ is equal to

If $u=\log \left(x^3+y^3+z^3-3 x y z\right)$,then $(x+y+z)(u_x+u_y+u_z)$ is equal to