Differentiate the following w.r.t. x or t or u as the case may be:
7. `y= (2x -3)/( 3x + 4)`

Answer

To differentiate the function \( y = \frac{2x - 3}{3x + 4} \) with respect to \( x \), we will use the quotient rule of differentiation. The quotient rule states that if you have a function in the form \( y = \frac{u}{v} \), then the derivative \( \frac{dy}{dx} \) is given by:

\[ \frac{dy}{dx} = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2} \]

where \( u = 2x - 3 \) and \( v = 3x + 4 \).

### Step-by-Step Solution:

1. **Identify \( u \) and \( v \)**:
- Let \( u = 2x - 3 \)
- Let \( v = 3x + 4 \)

2. **Differentiate \( u \) and \( v \)**:
- \( \frac{du}{dx} = \frac{d}{dx}(2x - 3) = 2 \)
- \( \frac{dv}{dx} = \frac{d}{dx}(3x + 4) = 3 \)

3. **Apply the Quotient Rule**:
- Using the quotient rule:
\[ \frac{dy}{dx} = \frac{(3x + 4)(2) - (2x - 3)(3)}{(3x + 4)^2} \]

4. **Simplify the numerator**:
- Calculate \( (3x + 4)(2) \):
\[ 2(3x + 4) = 6x + 8 \]
- Calculate \( (2x - 3)(3) \):
\[ 3(2x - 3) = 6x - 9 \]
- Now substitute back into the equation:
\[ \frac{dy}{dx} = \frac{(6x + 8) - (6x - 9)}{(3x + 4)^2} \]

5. **Combine like terms in the numerator**:
- Simplifying the numerator:
\[ (6x + 8) - (6x - 9) = 6x + 8 - 6x + 9 = 17 \]

6. **Final expression**:
- Substitute back into the derivative:
\[ \frac{dy}{dx} = \frac{17}{(3x + 4)^2} \]

### Final Answer:
\[ \frac{dy}{dx} = \frac{17}{(3x + 4)^2} \]