\[\frac{\frac{1}{\sqrt{x+\Delta x}}-\frac{1}{\sqrt{x}}}{\Delta x}=\frac{\sqrt{x}-\sqrt{x+\Delta x}}{\sqrt{x(x+\Delta x)}\Delta x}=\frac{\sqrt{x}-\sqrt{x+\Delta x}}{\sqrt{x(x+\Delta x)}(x-x+\Delta x)}=-\frac{1}{\sqrt{x(x+\Delta x)}(\sqrt{x}+\sqrt{x+\Delta x})}\]
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