Answer
$${G_s}\left( {s,t} \right) = \frac{{\sqrt {st} \left( {t - s} \right)}}{{2s{{\left( {s + t} \right)}^2}}}{\text{ and }}{G_t}\left( {s,t} \right) = \frac{{\sqrt {st} \left( {t - s} \right)}}{{2t{{\left( {s + t} \right)}^2}}}$$
Work Step by Step
$$\eqalign{
& G\left( {s,t} \right) = \frac{{\sqrt {st} }}{{s + t}} \cr
& {\text{Calculate }}{G_s}\left( {s,t} \right),{\text{ treat }}t{\text{ as a constant}} \cr
& {G_s}\left( {s,t} \right) = \frac{\partial }{{\partial s}}\left[ {\frac{{\sqrt {st} }}{{s + t}}} \right] \cr
& {G_s}\left( {s,t} \right) = \frac{{\left( {s + t} \right)\left( {\frac{t}{{2\sqrt {st} }}} \right) - \sqrt {st} }}{{{{\left( {s + t} \right)}^2}}} \cr
& {\text{Simplifying}} \cr
& {G_s}\left( {s,t} \right) = \frac{{\frac{{t\left( {s + t} \right) - 2{{\left( {\sqrt {st} } \right)}^2}}}{{2\sqrt {st} }}}}{{{{\left( {s + t} \right)}^2}}} \cr
& {G_s}\left( {s,t} \right) = \frac{{st + {t^2} - 2st}}{{2\sqrt {st} {{\left( {s + t} \right)}^2}}} \cr
& {G_s}\left( {s,t} \right) = \frac{{{t^2} - st}}{{2\sqrt {st} {{\left( {s + t} \right)}^2}}} \cr
& {G_s}\left( {s,t} \right) = \frac{{t\left( {t - s} \right)}}{{2\sqrt {st} {{\left( {s + t} \right)}^2}}} \cr
& {G_s}\left( {s,t} \right) = \frac{{\sqrt {st} \left( {t - s} \right)}}{{2s{{\left( {s + t} \right)}^2}}} \cr
& \cr
& {\text{Calculate }}{G_t}\left( {s,t} \right),{\text{ treat }}s{\text{ as a constant}} \cr
& {G_t}\left( {s,t} \right) = \frac{\partial }{{\partial s}}\left[ {\frac{{\sqrt {st} }}{{s + t}}} \right] \cr
& {G_t}\left( {s,t} \right) = \frac{{\left( {s + t} \right)\left( {\frac{s}{{2\sqrt {st} }}} \right) - \sqrt {st} }}{{{{\left( {s + t} \right)}^2}}} \cr
& {\text{Simplifying}} \cr
& {G_t}\left( {s,t} \right) = \frac{{\frac{{s\left( {s + t} \right) - 2{{\left( {\sqrt {st} } \right)}^2}}}{{2\sqrt {st} }}}}{{{{\left( {s + t} \right)}^2}}} \cr
& {G_t}\left( {s,t} \right) = \frac{{{s^2} + st - 2st}}{{2\sqrt {st} {{\left( {s + t} \right)}^2}}} \cr
& {G_t}\left( {s,t} \right) = \frac{{{s^2} - st}}{{2\sqrt {st} {{\left( {s + t} \right)}^2}}} \cr
& {G_t}\left( {s,t} \right) = \frac{{s\left( {t - s} \right)}}{{2\sqrt {st} {{\left( {s + t} \right)}^2}}} \cr
& {G_t}\left( {s,t} \right) = \frac{{\sqrt {st} \left( {t - s} \right)}}{{2t{{\left( {s + t} \right)}^2}}} \cr} $$
