\(=\left(\frac{\sqrt{y}}{\sqrt{x}\left(\sqrt{x}+\sqrt{y}\right)}-\frac{\sqrt{x}}{\sqrt{y}\left(\sqrt{x}+\sqrt{y}\right)}\right)\cdot\frac{\left(\sqrt{x}+\sqrt{y}\right)\left(\sqrt{x}-\sqrt{y}\right)}{2\sqrt{xy}}\)
\(=\left(\frac{y-x}{\sqrt{xy}\left(\sqrt{x}+\sqrt{y}\right)}\right)\cdot\frac{\left(\sqrt{x}+\sqrt{y}\right)\left(\sqrt{x}-\sqrt{y}\right)}{2\sqrt{xy}}\)
\(=\frac{\left(y-x\right)\left(\sqrt{x}-\sqrt{y}\right)}{2xy}\)
\(\left(\frac{\sqrt{y}}{x+\sqrt{xy}}-\frac{\sqrt{y}}{x-\sqrt{xy}}\right).\frac{x-y}{2\sqrt{xy}}\)
\(=\frac{x-y}{2\sqrt{xy}}.\left[-\frac{2y\sqrt{x}}{\left(x+\sqrt{yx}\right)\left(x-\sqrt{yx}\right)}\right]\)
\(=-\frac{2y\sqrt{x}}{\left(x+\sqrt{xy}\right)\left(x-\sqrt{xy}\right)}.\frac{x-y}{2\sqrt{xy}}\)
\(=-\frac{2y\sqrt{x}.\left(x-y\right)}{\left(x+\sqrt{xy}\right)\left(x-\sqrt{xy}\right).2\sqrt{xy}}\)
\(=-\frac{y\sqrt{x}\left(x-y\right)}{\left(x+\sqrt{xy}\right)\left(x-\sqrt{xy}\right).\sqrt{xy}}\)
\(=-\frac{\sqrt{y}\left(x-y\right)}{\left(x+\sqrt{yx}\right)\left(x-\sqrt{yx}\right)}\)
\(=-\frac{\sqrt{y}}{x}\)
