Find the derivatives of the following :
`f(x)=(3x+2)/((x+5)(2x+1)+3)`

Answer

To find the derivative of the function \( f(x) = \frac{3x + 2}{(x + 5)(2x + 1) + 3} \), we will use the quotient rule for differentiation. The quotient rule states that if you have a function in the form \( \frac{u}{v} \), then the derivative \( f'(x) \) is given by:

\[ f'(x) = \frac{u'v - uv'}{v^2} \]

where \( u = 3x + 2 \) and \( v = (x + 5)(2x + 1) + 3 \).

### Step 1: Identify \( u \) and \( v \)
- \( u = 3x + 2 \)
- \( v = (x + 5)(2x + 1) + 3 \)

### Step 2: Differentiate \( u \)
The derivative of \( u \):
\[ u' = \frac{d}{dx}(3x + 2) = 3 \]

### Step 3: Simplify and Differentiate \( v \)
First, we need to simplify \( v \):
\[ v = (x + 5)(2x + 1) + 3 \]
Expanding \( (x + 5)(2x + 1) \):
\[ = 2x^2 + x + 10x + 5 = 2x^2 + 11x + 5 \]
Now add 3:
\[ v = 2x^2 + 11x + 5 + 3 = 2x^2 + 11x + 8 \]

Now, differentiate \( v \):
\[ v' = \frac{d}{dx}(2x^2 + 11x + 8) = 4x + 11 \]

### Step 4: Apply the Quotient Rule
Now we can apply the quotient rule:
\[ f'(x) = \frac{(3)(2x^2 + 11x + 8) - (3x + 2)(4x + 11)}{(2x^2 + 11x + 8)^2} \]

### Step 5: Simplify the Numerator
Now we will simplify the numerator:
1. Expand \( 3(2x^2 + 11x + 8) \):
\[ = 6x^2 + 33x + 24 \]
2. Expand \( (3x + 2)(4x + 11) \):
\[ = 12x^2 + 33x + 8 \]
3. Combine the two results:
\[ 6x^2 + 33x + 24 - (12x^2 + 33x + 8) = 6x^2 + 33x + 24 - 12x^2 - 33x - 8 \]
\[ = -6x^2 + 16 \]

### Step 6: Write the Final Derivative
Thus, the derivative \( f'(x) \) is:
\[ f'(x) = \frac{-6x^2 + 16}{(2x^2 + 11x + 8)^2} \]

### Final Answer
\[ f'(x) = \frac{-6x^2 + 16}{(2x^2 + 11x + 8)^2} \]