Let $f(x,y)=\sqrt[\phantom{2}]{x^2+y^2}.$

$($a$)(7$ points$)$ Suppose $x(t)=t^3+1$ and $y(t)=sin(2t).$ Write down the appropriate chain rule for $\frac{df}{dt},$ and use this to evaluate $\frac{df}{dt}$ at $t=0.$

$($b$)(7$ points$)$ Suppose $x(u,v)=\frac{u}{v}$ and $y(u,v)=ln(2u+3v).$ Write down the appropriate chain rule for $\frac{\text{d}\text{e}\text{l}\text{f}}{\text{d}\text{e}\text{l}\text{u}},$ and use this to evaluate $\frac{\text{d}\text{e}\text{l}\text{f}}{\text{d}\text{e}\text{l}\text{u}}$ at $u=2,v=-1.$