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

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

$(Use$ 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 $\frac{\text{d}\text{e}\text{l}\text{x}}{\text{d}\text{e}\text{l}\text{x}},\frac{\text{d}\text{e}\text{l}\text{y}}{\text{d}\text{e}\text{l}\text{x}},\frac{\text{d}\text{e}\text{l}\text{x}}{\text{d}\text{e}\text{l}\text{x}}.$

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

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

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

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

$(c)$ Compute $\frac{df}{dx}$ using the Chain Rule.

$\frac{\text{d}\text{e}\text{l}\text{f}}{\text{d}\text{e}\text{l}\text{s}}=\frac{\text{d}\text{e}\text{l}\text{f}}{\text{d}\text{e}\text{l}\text{x}}\frac{\text{d}\text{e}\text{l}\text{x}}{\text{d}\text{e}\text{l}\text{s}}+\frac{\text{d}\text{e}\text{l}\text{f}}{\text{d}\text{e}\text{l}\text{y}}\frac{\text{d}\text{e}\text{l}\text{y}}{\text{d}\text{e}\text{l}\text{s}}+\frac{\text{d}\text{e}\text{l}\text{f}}{\text{d}\text{e}\text{l}\text{z}}\frac{\text{d}\text{e}\text{l}\text{z}}{\text{d}\text{e}\text{l}\text{s}}$

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

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