Let $f(x,y,z)=x^3{y}^4{z}^3$ and $x=s^5,y=s{t}^2,$ and $z=s^{5}t.$

$($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.$)$

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

$$

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

$$

$$

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

$$

$($b$)$ Calculate $\frac{\text{d}\text{e}\text{l}\text{x}}{\text{d}\text{e}\text{l}\text{s}},\frac{\text{d}\text{e}\text{l}\text{y}}{\text{d}\text{e}\text{l}\text{s}},\frac{\text{d}\text{e}\text{l}\text{z}}{\text{d}\text{e}\text{l}\text{s}}.$