If $f(x, y) = \frac{\cos(x - 4y)}{\cos(x + 4y)}$, then $\left. \frac{\partial f}{\partial x} \right|_{y = \frac{x}{4}}$ is equal to:

If $u = \sin^{-1} \sqrt{\frac{x^2 + y^2}{x + y}}$,then $x u_x + y u_y$ is

If $u = (x^2 + y^2 + z^2)^{3/2}$,then $\left( \frac{\partial u}{\partial x} \right)^2 + \left( \frac{\partial u}{\partial y} \right)^2 + \left( \frac{\partial u}{\partial z} \right)^2 = $

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 $u = \log_e(x^2 + y^2) + \tan^{-1}\left(\frac{y}{x}\right)$,then $\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = $

If $u = \frac{x + y}{x - y}$,then $\frac{\partial u}{\partial x} + \frac{\partial u}{\partial y} = $