If $u(x,y) = y \log x + x \log y$,then ${u_x}{u_y} - {u_x} \log x - {u_y} \log y + \log x \log 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 $u = x^2 + y^2$ and $x = s + 3t, y = 2s - t,$ then $\frac{d^2u}{ds^2} = $

If $z = \sin^{-1}\left( \frac{x+y}{\sqrt{x} + \sqrt{y}} \right)$,then $x\frac{\partial z}{\partial x} + y\frac{\partial z}{\partial y}$ is equal to

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

If $u = \sin^{-1}\left(\frac{y}{x}\right)$,then $\frac{\partial u}{\partial x}$ is equal to