To differentiate \(\frac{3x+2}{2x^2+11x+8}\), we'll use the **quotient rule**. The quotient rule is given by:

\[
\frac{d}{dx} \left( \frac{u(x)}{v(x)} \right) = \frac{v(x) \cdot u'(x) - u(x) \cdot v'(x)}{(v(x))^2}
\]

Where \( u(x) = 3x + 2 \) and \( v(x) = 2x^2 + 11x + 8 \).

### Step 1: Differentiate \( u(x) \) and \( v(x) \)
- \( u'(x) = 3 \)
- \( v'(x) = 4x + 11 \)

### Step 2: Apply the quotient rule
Using the formula:

\[
\frac{d}{dx} \left( \frac{3x+2}{2x^2+11x+8} \right) = \frac{(2x^2 + 11x + 8)(3) - (3x+2)(4x+11)}{(2x^2 + 11x + 8)^2}
\]

### Step 3: Simplify the numerator
Now, we expand both terms in the numerator:

1. First term: \( (2x^2 + 11x + 8)(3) = 6x^2 + 33x + 24 \)
2. Second term: \( (3x + 2)(4x + 11) = 12x^2 + 33x + 22 \)

Now, subtract the second term from the first:

\[
6x^2 + 33x + 24 - (12x^2 + 33x + 22) = -6x^2 + 2
\]

### Step 4: Write the final derivative
Thus, the derivative is:

\[
\frac{d}{dx} \left( \frac{3x+2}{2x^2+11x+8} \right) = \frac{-6x^2 + 2}{(2x^2 + 11x + 8)^2}
\]

Would you like a more detailed explanation on any of the steps or have further questions?

Here are 5 related questions:
1. How do you apply the product rule in differentiation?
2. What is the chain rule in calculus?
3. How do you find the derivative of a logarithmic function?
4. How do you differentiate implicit functions?
5. What is the significance of critical points in a function?

**Tip:** Always remember to check if simplification is possible after applying differentiation rules.

Learn how to differentiate the function (3x+2)/(2x^2+11x+8) using the quotient rule. This detailed step-by-step solution explains the key concepts of differentiation, including the application of the quotient rule and algebraic simplification. Suitable for students in Grades 11-12.

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