Differentiate the following functions w.r.t. x.
`f(x) = (3x +4)/( 4x +3) (x ne (-3)/(4) )`

Answer

To differentiate the function \( f(x) = \frac{3x + 4}{4x + 3} \) with respect to \( x \), we will use the quotient rule. The quotient rule states that if you have a function in the form \( \frac{u}{v} \), then the derivative is given by:

\[ \frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2} \]

where \( u = 3x + 4 \) and \( v = 4x + 3 \).

### Step-by-Step Solution:

1. **Identify \( u \) and \( v \)**:
- \( u = 3x + 4 \)
- \( v = 4x + 3 \)

2. **Differentiate \( u \) and \( v \)**:
- \( \frac{du}{dx} = 3 \) (since the derivative of \( 3x \) is \( 3 \) and the derivative of a constant \( 4 \) is \( 0 \))
- \( \frac{dv}{dx} = 4 \) (since the derivative of \( 4x \) is \( 4 \) and the derivative of a constant \( 3 \) is \( 0 \))

3. **Apply the Quotient Rule**:
\[ \frac{d}{dx}\left(\frac{3x + 4}{4x + 3}\right) = \frac{(4x + 3)(3) - (3x + 4)(4)}{(4x + 3)^2} \]

4. **Simplify the numerator**:
- Expand the terms:
\[ = \frac{(12x + 9) - (12x + 16)}{(4x + 3)^2} \]
- Combine like terms:
\[ = \frac{12x + 9 - 12x - 16}{(4x + 3)^2} \]
- This simplifies to:
\[ = \frac{-7}{(4x + 3)^2} \]

5. **Final Result**:
\[ f'(x) = \frac{-7}{(4x + 3)^2} \]