$(15\%\%)$ Compute the first derivative of the following expression:

$f(x)=\frac{\sqrt[\phantom{2}]{x}}{ln(x^2+e^{2x})}$

Discuss if the above expression defines a function.

$)$ Compute the partial derivatives $f_x$ and $f_{yx}$ of the following function:

$f(x,y)=\frac{(x+2)}{(y-x)}$

$(15\%)$ Solve this optimization problem indicating if the solution$($s$)$ is $($are$)$ a maximum or a minimum:

$f(x,y)=x^2+y^2+2x-2y+1$ subject $to{x}^2+y^2=2$

$(15\%)$ Consider the following system:

$x+2z-3=0$

$ax+y=2$

$2y+3z-3=0$

a$)$ Find the conditions by which the above system will admit a unique solution;

b$)$ Solve the system using Cramer's rule for $a=-1.$