To simplify the difference quotient

$\frac{\left( \frac{1}{\sqrt{a + h}} - \frac{1}{\sqrt{a}} \right)}{h}$

we will follow these steps:

Step 1: Combine the terms in the numerator

We have two fractions in the numerator:

$\frac{1}{\sqrt{a + h}} - \frac{1}{\sqrt{a}}$

To combine them, find a common denominator. The common denominator will be $\sqrt{a + h} \cdot \sqrt{a}$. So:

$\frac{1}{\sqrt{a + h}} - \frac{1}{\sqrt{a}} = \frac{\sqrt{a} - \sqrt{a + h}}{\sqrt{a} \cdot \sqrt{a + h}}$

Now the difference quotient becomes:

$\frac{\frac{\sqrt{a} - \sqrt{a + h}}{\sqrt{a} \cdot \sqrt{a + h}}}{h}$

Step 2: Simplify the complex fraction

We now have a complex fraction, so we can rewrite it as:

$\frac{\sqrt{a} - \sqrt{a + h}}{h \cdot \sqrt{a} \cdot \sqrt{a + h}}$

Step 3: Rationalize the numerator

To simplify further, we need to eliminate the square roots from the numerator by multiplying the numerator and denominator by the conjugate of the numerator, $\sqrt{a} + \sqrt{a + h}$:

$\frac{\sqrt{a} - \sqrt{a + h}}{h \cdot \sqrt{a} \cdot \sqrt{a + h}} \cdot \frac{\sqrt{a} + \sqrt{a + h}}{\sqrt{a} + \sqrt{a + h}}$

This multiplication simplifies the numerator using the difference of squares formula:

$(\sqrt{a} - \sqrt{a + h})(\sqrt{a} + \sqrt{a + h}) = a - (a + h) = -h$

Now the expression becomes:

$\frac{-h}{h \cdot \sqrt{a} \cdot \sqrt{a + h} \cdot (\sqrt{a} + \sqrt{a + h})}$

Step 4: Cancel out the $h$

Since there is an $h$ in both the numerator and denominator, we can cancel them:

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

Final Answer:

The simplified form of the difference quotient is:

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

Would you like further details or clarification on any step?

Here are 5 related questions to deepen your understanding:

1. How does the difference quotient relate to the definition of the derivative?
2. What is the purpose of rationalizing the numerator in such expressions?
3. Can the difference quotient be used to find the derivative of other functions with square roots?
4. What happens if $h \to 0$ in this expression? How does it relate to limits?
5. How do you apply the difference of squares formula in similar problems?

Tip: Always look for opportunities to factor or rationalize expressions when simplifying complex fractions involving square roots.