Answer
The simplified form of the expression $\frac{\frac{1}{\sqrt{x+h}}-\frac{1}{\sqrt{x}}}{h}$ is $\frac{\sqrt{x}-\sqrt{x+h}}{h\sqrt{x\left( x+h \right)}}$.
Work Step by Step
Consider the provided expression, $\frac{\frac{1}{\sqrt{x+h}}-\frac{1}{\sqrt{x}}}{h}$
Multiply the numerator and denominator by $\sqrt{x}\sqrt{x+h}$.
Therefore,
$\frac{\frac{1}{\sqrt{x+h}}-\frac{1}{\sqrt{x}}}{h}=\frac{\frac{1}{\sqrt{x+h}}-\frac{1}{\sqrt{x}}}{h}\cdot \frac{\sqrt{x}\sqrt{x+h}}{\sqrt{x}\sqrt{x+h}}$
Apply the distributive property in the numerator: $a\left( b+c \right)=ab+ac$
Therefore,
$$ $\begin{align}
& \frac{\frac{1}{\sqrt{x+h}}-\frac{1}{\sqrt{x}}}{h}=\frac{\frac{1}{\sqrt{x+h}}\sqrt{x}\sqrt{x+h}-\frac{1}{\sqrt{x}}\sqrt{x}\sqrt{x+h}}{h\sqrt{x}\sqrt{x+h}} \\
& =\frac{\sqrt{x}-\sqrt{x+h}}{h\sqrt{x}\sqrt{x+h}} \\
& =\frac{\sqrt{x}+\sqrt{x+h}}{h\sqrt{x\left( x+h \right)}}
\end{align}$ $$
Here, $h\ne 0$
Therefore, the simplified form of the expression $\frac{\frac{1}{\sqrt{x+h}}-\frac{1}{\sqrt{x}}}{h}$ is $\frac{\sqrt{x}-\sqrt{x+h}}{h\sqrt{x\left( x+h \right)}}$.
