Answer
$-\frac{1}{\sqrt{x}\sqrt{x+h}(\sqrt{x}+\sqrt{x+h})}$
Work Step by Step
First we form a common denominator to combine the top fractions. Then we multiply by $\sqrt{x}+\sqrt{x+h}$ and use the fact that $(a-b)(a+b)=a^b-b^2$:
$\displaystyle \frac{\frac{1}{\sqrt{x+h}}-\frac{1}{\sqrt{x}}}{h}=\frac{\frac{\sqrt{x}}{\sqrt{x}\sqrt{x+h}}-\frac{\sqrt{x+h}}{\sqrt{x+h}\sqrt{x}}}{h}=\frac{\sqrt{x}-\sqrt{x+h}}{h\sqrt{x}\sqrt{x+h}}\cdot\frac{\sqrt{x}+\sqrt{x+h}}{\sqrt{x}+\sqrt{x+h}}=\frac{x-(x+h)}{h\sqrt{x}\sqrt{x+h}(\sqrt{x}+\sqrt{x+h})}=-\frac{1}{\sqrt{x}\sqrt{x+h}(\sqrt{x}+\sqrt{x+h})}$
