To find the derivative of the function $f(x) = \frac{1}{3x+2}$ using the limit definition of the derivative, we proceed as follows:

Step 1: Write the Limit Definition of the Derivative

The derivative of $f(x)$ at any point $x$ is given by:

$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$

Step 2: Substitute the Function into the Definition

We know $f(x) = \frac{1}{3x+2}$. So, we first need to find $f(x+h)$:

$f(x+h) = \frac{1}{3(x+h) + 2} = \frac{1}{3x + 3h + 2}$

Now, substitute $f(x+h)$ and $f(x)$ into the limit definition:

$f'(x) = \lim_{h \to 0} \frac{\frac{1}{3x + 3h + 2} - \frac{1}{3x + 2}}{h}$

Step 3: Simplify the Expression

To simplify the expression, combine the two fractions in the numerator:

$f'(x) = \lim_{h \to 0} \frac{\frac{(3x + 2) - (3x + 3h + 2)}{(3x + 3h + 2)(3x + 2)}}{h}$

Simplify the numerator:

$f'(x) = \lim_{h \to 0} \frac{\frac{3x + 2 - 3x - 3h - 2}{(3x + 3h + 2)(3x + 2)}}{h} = \lim_{h \to 0} \frac{\frac{-3h}{(3x + 3h + 2)(3x + 2)}}{h}$

Simplify further by canceling $h$ in the numerator and denominator:

$f'(x) = \lim_{h \to 0} \frac{-3}{(3x + 3h + 2)(3x + 2)}$

Step 4: Evaluate the Limit

Finally, take the limit as $h$ approaches 0:

$f'(x) = \frac{-3}{(3x + 2)(3x + 2)} = \frac{-3}{(3x + 2)^2}$

So, the derivative of $f(x) = \frac{1}{3x+2}$ is:

$f'(x) = \frac{-3}{(3x + 2)^2}$

Would you like more details, or do you have any questions?

Here are 8 related questions you might consider:

1. How would the derivative change if $f(x) = \frac{1}{5x+3}$?
2. What is the derivative of $f(x) = \frac{1}{3x+2}$ using the chain rule?
3. How can you apply the limit definition to find the second derivative of $f(x) = \frac{1}{3x+2}$?
4. Can you find the derivative of $f(x) = \frac{1}{3x+2}$ using the quotient rule?
5. How does the concept of limits apply to the continuity of $f(x) = \frac{1}{3x+2}$?
6. What is the significance of the negative sign in the derivative $f'(x) = \frac{-3}{(3x+2)^2}$?
7. How would you interpret the graph of the derivative function $f'(x)$?
8. Can you find the derivative of a similar function, such as $f(x) = \frac{1}{x^2 + 1}$?

Tip: When using the limit definition of the derivative, always ensure to simplify the expression before taking the limit for easier calculation.