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

$($a$)$ Calculate the primary derivatives

$\frac{\text{d}\text{e}\text{l}\text{f}}{\text{d}\text{e}\text{l}\text{x}}=3x^2{y}^3+3x^2$

$\frac{\text{d}\text{e}\text{l}\text{f}}{\text{d}\text{e}\text{l}\text{y}}=3x^3{y}^2$

$\frac{\text{d}\text{e}\text{l}\text{f}}{\text{d}\text{e}\text{l}\text{z}}=3z^2$

$($b$)$ Calculate

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

$\frac{\text{d}\text{e}\text{l}\text{y}}{\text{d}\text{e}\text{l}\text{s}}=3s^2{t}^3$

$\frac{\text{d}\text{e}\text{l}\text{z}}{\text{d}\text{e}\text{l}\text{s}}=3s^{2}t$

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

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

In $($c$)$ express your answer in terms of the independent variables $t,s$