Use the limit definition of the derivative to find f'(x) if f(x) = 1/(3x+2)

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.

Learn how to find the derivative of the function f(x) = 1/(3x+2) using the limit definition of the derivative. This problem involves calculus concepts such as derivatives and the limit definition, suitable for advanced high school or college students.

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