Problem $1.(40$ points$)$ Consider the function $f(x,y,z)=x{y}^{-1}-yz.$

$($a$)$ Find the direction in which $f(x,y,z)$ increases most rapidly at the point $P_0(4,1,1),$ then find the derivative of $f$ at $P_0$ in this direction.

$($b$)$ Evaluate the following limit:

$\underset{(x,y,z)(4,1,1)}{lim}f(x,y,z)$

$($c$)$ Suppose $x=s+t,y=2s-t,z=\frac{1}{s}+\frac{1}{t}.$ Find $\frac{\text{d}\text{e}\text{l}\text{f}}{\text{d}\text{e}\text{l}\text{s}}$ and $\frac{\text{d}\text{e}\text{l}\text{f}}{\text{d}\text{e}\text{l}\text{t}}$ when $(s,t)=(1,1).$

$($d$)$ Use implicit differentiation to find $\frac{\text{d}\text{e}\text{l}\text{z}}{\text{d}\text{e}\text{l}\text{x}}$ at the point $(4,1,1)$ if the level surface $f(x,y,z)=1$ defines $z$ as a function of two independent variables $x$ and $y.$