`(2x +3)/(3x+2)`

Explanation

To differentiate the function \( f(x) = \frac{2x + 3}{3x + 2} \) with respect to \( x \), we will use the quotient rule. The quotient rule states that if you have a function in the form of \( \frac{U}{V} \), then the derivative is given by:

\[ \frac{d}{dx}\left(\frac{U}{V}\right) = \frac{V \cdot \frac{dU}{dx} - U \cdot \frac{dV}{dx}}{V^2} \]

### Step 1: Identify \( U \) and \( V \)
Let:
- \( U = 2x + 3 \)
- \( V = 3x + 2 \)

### Step 2: Differentiate \( U \) and \( V \)
Now, we need to find the derivatives of \( U \) and \( V \):
- \( \frac{dU}{dx} = 2 \)
- \( \frac{dV}{dx} = 3 \)

### Step 3: Apply the Quotient Rule
Now we apply the quotient rule:

\[ \frac{d}{dx}\left(\frac{U}{V}\right) = \frac{(3x + 2)(2) - (2x + 3)(3)}{(3x + 2)^2} \]

### Step 4: Simplify the numerator
Now we simplify the numerator:

\[ = \frac{(6x + 4) - (6x + 9)}{(3x + 2)^2} \]

### Step 5: Combine like terms
Now combine like terms in the numerator:

\[ = \frac{6x + 4 - 6x - 9}{(3x + 2)^2} \]
\[ = \frac{-5}{(3x + 2)^2} \]

### Final Answer
Thus, the derivative of \( f(x) = \frac{2x + 3}{3x + 2} \) is:

\[ f'(x) = \frac{-5}{(3x + 2)^2} \]

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