$(20$ points$)$ Suppose $w=x{y}^2+x^{2}z+xy{z}^2,x=st+u,y=s-u^2,$ and $z=t^2-su.$

$(a)$ Draw a dependency diagram and write a Chain Rule formula for the derivative $\frac{\text{d}\text{e}\text{l}\text{w}}{\text{d}\text{e}\text{l}\text{s}}.$

$(b)$ Use the chain rule $to$ compute $\frac{\text{d}\text{e}\text{l}\text{w}}{\text{d}\text{e}\text{l}\text{s}}$ when $s=1,t=1,u=0.$