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The image shows a mathematical problem asking to find the difference quotient \(\frac{f(x+h) - f(x)}{h}\) for the given function \( f(x) = \frac{1}{\sqrt{3x - 2}} \) as \( h \to 0 \).

### Solution:
The difference quotient is a fundamental part of finding the derivative. Let's proceed step-by-step.

1. **Function \( f(x) \):**
   \[
   f(x) = \frac{1}{\sqrt{3x - 2}}
   \]

2. **Function \( f(x+h) \):**
   Substitute \( x + h \) into the function:
   \[
   f(x + h) = \frac{1}{\sqrt{3(x + h) - 2}} = \frac{1}{\sqrt{3x + 3h - 2}}
   \]

3. **Difference Quotient:**
   Now, apply the formula for the difference quotient:
   \[
   \frac{f(x + h) - f(x)}{h} = \frac{\frac{1}{\sqrt{3x + 3h - 2}} - \frac{1}{\sqrt{3x - 2}}}{h}
   \]
   To simplify this, combine the terms in the numerator:
   \[
   \frac{f(x + h) - f(x)}{h} = \frac{\sqrt{3x - 2} - \sqrt{3x + 3h - 2}}{h \cdot \sqrt{3x + 3h - 2} \cdot \sqrt{3x - 2}}
   \]
   The next step is to simplify the numerator using a technique called "rationalizing the numerator," multiplying both the numerator and the denominator by the conjugate of the numerator:
   \[
   \text{Conjugate} = \sqrt{3x - 2} + \sqrt{3x + 3h - 2}
   \]
   This gives:
   \[
   \frac{\left( \sqrt{3x - 2} - \sqrt{3x + 3h - 2} \right) \left( \sqrt{3x - 2} + \sqrt{3x + 3h - 2} \right)}{h \cdot \left( \sqrt{3x + 3h - 2} \cdot \sqrt{3x - 2} \right) \left( \sqrt{3x - 2} + \sqrt{3x + 3h - 2} \right)}
   \]
   The numerator becomes:
   \[
   (3x - 2) - (3x + 3h - 2) = -3h
   \]
   Now the expression is:
   \[
   \frac{-3h}{h \cdot \left( \sqrt{3x + 3h - 2} \cdot \sqrt{3x - 2} \right) \left( \sqrt{3x - 2} + \sqrt{3x + 3h - 2} \right)}
   \]
   Simplifying:
   \[
   \frac{-3}{\sqrt{3x + 3h - 2} \cdot \sqrt{3x - 2} \cdot \left( \sqrt{3x - 2} + \sqrt{3x + 3h - 2} \right)}
   \]

4. **Taking the Limit as \( h \to 0 \):**
   As \( h \to 0 \), \( \sqrt{3x + 3h - 2} \to \sqrt{3x - 2} \). So the expression simplifies to:
   \[
   \frac{-3}{2 \cdot \left( \sqrt{3x - 2} \right)^3}
   \]
   Therefore, the derivative of the function \( f(x) \) is:
   \[
   f'(x) = \frac{-3}{2(3x - 2)^{3/2}}
   \]

Would you like more details or have any questions?

Here are some related questions you might find interesting:
1. What is the geometric interpretation of a derivative?
2. How does the difference quotient relate to the concept of a tangent line?
3. Can you apply this same process to a polynomial function?
4. What is the importance of rationalizing the numerator in calculus?
5. How would the derivative change if the function were \( f(x) = \frac{1}{\sqrt{ax - b}} \)?

**Tip:** Rationalizing the numerator is an essential algebraic technique to simplify expressions, especially when finding limits and derivatives.

Find \(\frac{f(x+h) - f(x)}{h}\) for the function \( f(x) = \frac{1}{\sqrt{3x - 2}} \) as \( h \to 0 \).

Learn how to calculate the difference quotient for the function f(x) = 1/√(3x - 2) and find its derivative using the limit definition. This step-by-step solution covers key calculus concepts such as limits and derivatives, suitable for advanced high school students.

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