Let $f$ be the $R^2-R$ function defined by

$f(x,y)=\frac{1}{x^2+y^2}$

and let $r_{?}$ be the $R-R^2$ function defined by

$r_t=(tcost,tsint)$

$($a$)$ Determine the composite function $f$@$r_{?}.($Simplify your answer$).$

$(2)$

$($b$)$ Determine $gra{d}_f(x,y)$ and $r_{{?}^{?}}(\text{'})(t).$

$(3)$

$($c$)$ Determine the derivative function by

i$.$ differentiating the expression obtained in $($a$)$

$(2)$

ii$.$ using the chain rule $($Theorem $7\mathrm{.}6\mathrm{.}1).$

$(2)$

Compare your answers.