Let $h(x,y)=e^{x+y}sin(xy),$ where $x=cos(u)$ and $y=sin(v).$

$($a$)$ Find $\frac{\text{d}\text{e}\text{l}\text{h}}{\text{d}\text{e}\text{l}\text{s}}$ using the chain rule.

$($b$)$ Evaluate $\frac{\text{d}\text{e}\text{l}\text{h}}{\text{d}\text{e}\text{l}\text{s}}$ at $s=1$ and $t=1.$

Let $h(x,y)=\frac{sin(x+y)}{x^2+y^2},$ where $x=\sqrt[\phantom{2}]{s^2+t^2}$ and $y=ta{n}^{-1}(\frac{t}{s}).$

$($a$)$ Find $\frac{\text{d}\text{e}\text{l}\text{h}}{\text{d}\text{e}\text{l}\text{u}}$ using the chain rule.

$($b$)$ Evaluate $\frac{\text{d}\text{e}\text{l}\text{h}}{\text{d}\text{e}\text{l}\text{u}}$ at $u=\frac{}{3}$ and $v=\frac{}{6}.$

Let $h(x,y)=xy{e}^{x-y},$ where $x=s^2+t$ and $y=s-t^2.$

$($a$)$ Find $\frac{\text{d}\text{e}\text{l}\text{h}}{\text{d}\text{e}\text{l}\text{s}}$ using the chain rule.

$($b$)$ Evaluate $\frac{\text{d}\text{e}\text{l}\text{h}}{\text{d}\text{e}\text{l}\text{s}}$ at $s=1$ and $t=1.$

Let $h(x,y,z)=(x+y{)}^2+z^3,$ where $x=s+2r+t,y=s^2-r,$ and $z=t-r^2.$

$($a$)$ Find $\frac{\text{d}\text{e}\text{l}\text{h}}{\text{d}\text{e}\text{l}\text{s}}$ using the chain rule.

$($b$)$ Evaluate $\frac{\text{d}\text{e}\text{l}\text{h}}{\text{d}\text{e}\text{l}\text{s}}$ at $s=1,t=1,$ and $r=2.$