Let $f(x,y,z)=x^2{y}^2+z^2$ and $x=s{t}^2,y=st,$ and $z=s^3{t}^3.$

$(a)$ Calculate the primary derivatives

$\frac{\text{d}\text{e}\text{l}\text{f}}{\text{d}\text{e}\text{l}\text{x}}=$

$\frac{\text{d}\text{e}\text{l}\text{f}}{\text{d}\text{e}\text{l}\text{y}}=$

$\frac{\text{d}\text{e}\text{l}\text{f}}{\text{d}\text{e}\text{l}\text{z}}=$

$(b)$ Calculate

$\frac{\text{d}\text{e}\text{l}\text{x}}{\text{d}\text{e}\text{l}\text{s}}=$

$\frac{\text{d}\text{e}\text{l}\text{y}}{\text{d}\text{e}\text{l}\text{s}}=$

$\frac{\text{d}\text{e}\text{l}\text{z}}{\text{d}\text{e}\text{l}\text{s}}=$

$(c)$ Use the Chain Rule $to$ compute

$\frac{\text{d}\text{e}\text{l}\text{f}}{\text{d}\text{e}\text{l}\text{s}}=$