\[ y'=\lim_{\Delta x\rightarrow 0}{\frac{1}{\Delta x}}\left[\frac{1}{\sqrt{x+\Delta x\;}}-\frac{1}{\sqrt{x\;}}\right]=\lim_{\Delta x\rightarrow 0}{\frac{1}{\Delta x}}\left[\frac{\sqrt{x\;}-\sqrt{x+\Delta x\;}}{\sqrt{x\;}\sqrt{x+\Delta x\;}}\right] \]

\[ \begin{align} y' &=\lim_{\Delta x\rightarrow 0}{\frac{1}{\Delta x}}\left[\frac{\sqrt{x\;}-\sqrt{x+\Delta x\;}}{\sqrt{x\;}\sqrt{x+\Delta x\;}}.\frac{\sqrt{x\;}+\sqrt{x+\Delta x\;}}{\sqrt{x\;}+\sqrt{x+\Delta x\;}}\right]=\\[5pt] &=\lim_{\Delta x\rightarrow 0}{\frac{1}{\Delta x}}\left[\frac{\left(\sqrt{x\;}\right)^{2}+\cancel{\sqrt{x\;}.\sqrt{x+\Delta x\;}}-\cancel{\sqrt{x+\Delta x\;}.\sqrt{x\;}}+\left(-\sqrt{x+\Delta x\;}\right).\sqrt{x+\Delta x\;}}{\sqrt{x\;}.\sqrt{x\;}\sqrt{x+\Delta x\;}+\sqrt{x\;}\sqrt{x+\Delta x\;}.\sqrt{x+\Delta x\;}}\right]=\\[5pt] &=\lim_{\Delta x\rightarrow 0}{\frac{1}{\Delta x}}\left[\frac{x-\left(x+\Delta x\right)}{x\sqrt{x+\Delta x\;}+\sqrt{x\;}(x+\Delta x)}\right]=\lim_{\Delta x\rightarrow 0}{\frac{1}{\Delta x}}\left[\frac{\cancel{x}-\cancel{x}-\Delta x}{x\sqrt{x+\Delta x\;}+\sqrt{x\;}(x+\Delta x)}\right]=\\[5pt] &=\lim_{\Delta x\rightarrow 0}{\frac{1}{\cancel{\Delta x}}}\left[\frac{-\cancel{\Delta x}}{x\sqrt{x+\Delta x\;}+\sqrt{x\;}(x+\Delta x)}\right]=\lim_{\Delta x\rightarrow 0}\left[-{\frac{1}{x\sqrt{x+\Delta x\;}+\sqrt{x\;}(x+\Delta x)}}\right]=\\[5pt] &=\lim_{\Delta x\rightarrow 0}\left[-{\frac{1}{x\sqrt{x+0\;}+\sqrt{x\;}(x+0)}}\right]=-{\frac{1}{x\sqrt{x\;}+x\sqrt{x\;}}}=-{\frac{1}{2x\sqrt{x\;}}} \end{align} \]