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

(a) Calculate the primary derivatives

$\frac{\partial{f}}{\partial{x}}=$
$\frac{\partial{f}}{\partial{y}}=$
$\frac{\partial{f}}{\partial{z}}=$

(b) Calculate

$\frac{\partial{x}}{\partial{s}}=$
$\frac{\partial{y}}{\partial{s}}=$
$\frac{\partial{z}}{\partial{s}}=$

(c) Use the Chain Rule to compute

$\frac{\partial{f}}{\partial{s}}=$
In (c) express your answer in terms of the independent variables $t,s$