`((2x^(2) - 4)/(3x^(2) + 7))`

Answer

To differentiate the function \( y = \frac{2x^2 - 4}{3x^2 + 7} \), we will use the quotient rule of differentiation. The quotient rule states that if you have a function in the form \( \frac{u}{v} \), where \( u \) and \( v \) are both functions of \( x \), then the derivative is given by:

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

### Step-by-Step Solution:

1. **Identify \( u \) and \( v \)**:
- Let \( u = 2x^2 - 4 \)
- Let \( v = 3x^2 + 7 \)

2. **Differentiate \( u \) and \( v \)**:
- \( \frac{du}{dx} = \frac{d}{dx}(2x^2 - 4) = 4x \)
- \( \frac{dv}{dx} = \frac{d}{dx}(3x^2 + 7) = 6x \)

3. **Apply the Quotient Rule**:
- Using the quotient rule:
\[ \frac{dy}{dx} = \frac{(3x^2 + 7)(4x) - (2x^2 - 4)(6x)}{(3x^2 + 7)^2} \]

4. **Expand the Numerator**:
- First term: \( (3x^2 + 7)(4x) = 12x^3 + 28x \)
- Second term: \( (2x^2 - 4)(6x) = 12x^3 - 24x \)
- Combine these:
\[ \text{Numerator} = 12x^3 + 28x - (12x^3 - 24x) = 12x^3 + 28x - 12x^3 + 24x = 52x \]

5. **Final Expression**:
- Substitute back into the derivative:
\[ \frac{dy}{dx} = \frac{52x}{(3x^2 + 7)^2} \]

### Final Answer:
\[ \frac{dy}{dx} = \frac{52x}{(3x^2 + 7)^2} \]