$($a$)$ Calculate $f_{232}(x,y,z)$ for the function $f(x,y,z)=e^{x-2y+3z}$

$($b$)$ Compute $\frac{dz}{dx}$ for $z=xln(xy)+y^3,y=cos(x^2+1)$

$($c$)$ Find and classify the critical points of the given function

$f(x,y)=3x^{2}y+y^3-3x^2-3y^2+2$