\(\left(\dfrac{1}{\sqrt{x}}-x\right):\left(\dfrac{\sqrt{x}-1}{\sqrt{x}}+\dfrac{1-\sqrt{x}}{x+\sqrt{x}}\right)\)
\(=\left(\dfrac{1}{\sqrt{x}}-\dfrac{x\sqrt{x}}{\sqrt{x}}\right):\left[\dfrac{\sqrt{x}-1}{\sqrt{x}}+\dfrac{1-\sqrt{x}}{\sqrt{x}\left(\sqrt{x}+1\right)}\right]\)
\(=\dfrac{1-x\sqrt{x}}{\sqrt{x}}:\left[\dfrac{\left(\sqrt{x}-1\right)\left(\sqrt{x}-1\right)}{\sqrt{x}\left(\sqrt{x}+1\right)}+\dfrac{1-\sqrt{x}}{\sqrt{x}\left(\sqrt{x}+1\right)}\right]\)
\(=\dfrac{\left(1-\sqrt{x}\right)\left(1+\sqrt{x}+x\right)}{\sqrt{x}}:\dfrac{x-1+1-\sqrt{x}}{\sqrt{x}\left(\sqrt{x}+1\right)}\)
\(=\dfrac{\left(1-\sqrt{x}\right)\left(x+\sqrt{x}+1\right)}{\sqrt{x}}\cdot\dfrac{\sqrt{x}\left(\sqrt{x}+1\right)}{x-\sqrt{x}}\)
\(=-\dfrac{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}{\sqrt{x}}\cdot\dfrac{\sqrt{x}\left(\sqrt{x}+1\right)}{\sqrt{x}\left(\sqrt{x}-1\right)}\)
\(=-\dfrac{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)\left(\sqrt{x}+1\right)}{\sqrt{x}\left(\sqrt{x}-1\right)}\)
\(=\dfrac{\left(x+\sqrt{x}+1\right)\left(\sqrt{x}+1\right)}{\sqrt{x}}\)
\(\left(\dfrac{1}{\sqrt{x}}-x\right):\left(\dfrac{\sqrt{x}-1}{\sqrt{x}}+\dfrac{1-\sqrt{x}}{x+\sqrt{x}}\right)=\dfrac{1-x\sqrt{x}}{\sqrt{x}}:\dfrac{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)+\left(1-\sqrt{x}\right)}{\sqrt{x}\left(\sqrt{x}+1\right)}=\dfrac{1-x\sqrt{x}}{\sqrt{x}}.\dfrac{\sqrt{x}\left(\sqrt{x}+1\right)}{x-\sqrt{x}}=\dfrac{\left(1-x\sqrt{x}\right)\left(\sqrt{x}+1\right)}{\sqrt{x}\left(\sqrt{x}+1\right)}=\dfrac{1-x\sqrt{x}}{\sqrt{x}}\)
