The given expression is:

$\frac{\frac{1}{\sqrt{x+h}} - \frac{1}{\sqrt{x}}}{h}.$

Let us simplify this step by step:

Step 1: Simplify the numerator

The numerator is:

$\frac{1}{\sqrt{x+h}} - \frac{1}{\sqrt{x}}.$

To simplify, find the common denominator:

$\text{Common denominator: } \sqrt{x+h} \cdot \sqrt{x}.$

So the numerator becomes:

$\frac{\sqrt{x} - \sqrt{x+h}}{\sqrt{x+h} \cdot \sqrt{x}}.$

Step 2: Write the entire expression

Now the entire fraction becomes:

$\frac{\frac{\sqrt{x} - \sqrt{x+h}}{\sqrt{x+h} \cdot \sqrt{x}}}{h}.$

This can be rewritten as:

$\frac{\sqrt{x} - \sqrt{x+h}}{h \cdot \sqrt{x+h} \cdot \sqrt{x}}.$

Step 3: Rationalize the numerator

To simplify further, rationalize the numerator $\sqrt{x} - \sqrt{x+h}$ by multiplying and dividing by its conjugate, $\sqrt{x} + \sqrt{x+h}$. This gives:

$\sqrt{x} - \sqrt{x+h} = \frac{(\sqrt{x} - \sqrt{x+h})(\sqrt{x} + \sqrt{x+h})}{\sqrt{x} + \sqrt{x+h}}.$

The numerator simplifies to:

$(\sqrt{x})^2 - (\sqrt{x+h})^2 = x - (x + h) = -h.$

Thus, the numerator becomes:

$\frac{-h}{\sqrt{x} + \sqrt{x+h}}.$

Step 4: Substitute back

The entire expression now becomes:

$\frac{\frac{-h}{\sqrt{x} + \sqrt{x+h}}}{h \cdot \sqrt{x+h} \cdot \sqrt{x}}.$

Cancel $h$ from the numerator and denominator:

$\frac{-1}{(\sqrt{x} + \sqrt{x+h}) \cdot \sqrt{x+h} \cdot \sqrt{x}}.$

Final Simplified Expression:

$\boxed{-\frac{1}{\sqrt{x} \cdot \sqrt{x+h} \cdot (\sqrt{x} + \sqrt{x+h})}}.$

Would you like further clarification or details about any step?

Related Questions:

1. What is the importance of rationalizing the numerator in such expressions?
2. How does this simplification apply to the concept of limits in calculus?
3. Can this expression be used to calculate a derivative? If so, how?
4. What happens if $h \to 0$ in the simplified expression?
5. How would the process change if the original numerator was $1/\sqrt{x+h} + 1/\sqrt{x}$?

Tip:

When simplifying fractions with radicals, always check for opportunities to rationalize either the numerator or denominator to make further simplification possible.