If $z=\sec (y-ax)+\tan (y+ax)$, then $\frac{\partial^2 z}{\partial x^2}-a^2 \frac{\partial^2 z}{\partial y^2}$ is equal to

If $u = \sin^{-1}\left(\frac{x}{y}\right) + \tan^{-1}\left(\frac{y}{x}\right)$, then the value of $x \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y}$ is

If $z = \frac{(x^4 + y^4)^{1/3}}{(x^3 + y^3)^{1/4}}$,then $x\frac{\partial z}{\partial x} + y\frac{\partial z}{\partial y} = $

$\begin{aligned} & f(x, y)=2(x-y)^2-x^4-y^4 \\ & \left|\left(f_{x x} f_{y y}-f_{x y}^2\right)\right|_{(0,0)} \end{aligned}$

If $z = \frac{y}{x} \left[ \sin \frac{x}{y} + \cos \left( 1 + \frac{y}{x} \right) \right]$, then $x \frac{\partial z}{\partial x}$ is equal to

If $u = u(x, y) = \sin(y + ax) - (y + ax)^2$, then which of the following is true?