Question 1027708
{{{r^2 + 2st^(1/2) = rt}}} is an implicit function of three variables.

(A) The partial derivative ds/dr is {{{2r + 2(ds/dr)t^(1/2) = t}}}

==> {{{ds/dr = (t - 2r)/ (2t^(1/2))}}}.

(B)  The partial derivative dr/dt is {{{2r(dr/dt) + 2s(1/2)(1/t^(1/2)) = (dr/dt)t + r}}} 

==> {{{2r(dr/dt) + s/t^(1/2) = (dr/dt)t + r}}} 

{{{(2r - t)(dr/dt) = r - s/t^(1/2)}}}

{{{dr/dt = (r - s/t^(1/2))/(2r - t)}}}

I leave parts (C) and (D) for you to practice on.