Answer
$\dfrac{1}{4}$ ; $\dfrac{-3}{4}$
Work Step by Step
Here,we have $F(x,y,z)= z^3-xy+yz+y^3-2=0$ and
$\dfrac{\partial z}{ \partial x}=-\dfrac{F_x}{F_z}$
and $\dfrac{\partial z}{ \partial x}=-(\dfrac{-y}{3z^2+y})=\dfrac{y}{y+3z^2}$
For the point $(1,1,1)$, we have
$\dfrac{\partial z}{ \partial x}=\dfrac{1}{4}$
$\dfrac{\partial z}{ \partial y}=-\dfrac{F_y}{F_z}$
$\implies \dfrac{\partial z}{ \partial y}=\dfrac{(x-z-3y^2)}{y+3z^2}$
Now, $\dfrac{\partial z}{ \partial y}(1,1,1)=\dfrac{1-1-3}{1+3}=\dfrac{-3}{4}$
