Let $f(x,y,z)=x^5{y}^5{z}^3$ and $x=s^3,y=s{t}^4,$ and $z=s^{4}t.$

$($a$)$ Calculate the primary derivatives $\frac{\text{d}\text{e}\text{l}\text{f}}{\text{d}\text{e}\text{l}\text{x}},\frac{\text{d}\text{e}\text{l}\text{f}}{\text{d}\text{e}\text{l}\text{y}},\frac{\text{d}\text{e}\text{l}\text{f}}{\text{d}\text{e}\text{l}\text{z}}.$

$($Use symbolic notation and fractions where needed.$)$