Differentiate the following w.r.t. x :
`((ax+b)(cx+d))/((ax-b)(cx-d)),xneb/a,d/c`

Answer

To differentiate the function \( y = \frac{(ax + b)(cx + d)}{(ax - b)(cx - d)} \) with respect to \( x \), we will use the quotient rule and the product rule.

### Step-by-Step Solution:

1. **Identify the Functions**:
Let
\[ u = (ax + b)(cx + d) \quad \text{and} \quad v = (ax - b)(cx - d) \]
Therefore, we can express \( y \) as:
\[ y = \frac{u}{v} \]

2. **Differentiate Using Quotient Rule**:
The quotient rule states that if \( y = \frac{u}{v} \), then:
\[ \frac{dy}{dx} = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2} \]

3. **Differentiate \( u \) and \( v \)**:
- **Differentiate \( u \)**:
\[ u = (ax + b)(cx + d) \]
Using the product rule:
\[ \frac{du}{dx} = (ax + b) \frac{d}{dx}(cx + d) + (cx + d) \frac{d}{dx}(ax + b) \]
\[ = (ax + b)(c) + (cx + d)(a) \]
\[ = c(ax + b) + a(cx + d) \]

- **Differentiate \( v \)**:
\[ v = (ax - b)(cx - d) \]
Again using the product rule:
\[ \frac{dv}{dx} = (ax - b) \frac{d}{dx}(cx - d) + (cx - d) \frac{d}{dx}(ax - b) \]
\[ = (ax - b)(c) + (cx - d)(a) \]
\[ = c(ax - b) + a(cx - d) \]

4. **Substitute \( \frac{du}{dx} \) and \( \frac{dv}{dx} \) into the Quotient Rule**:
Now substituting back into the quotient rule formula:
\[ \frac{dy}{dx} = \frac{v(c(ax + b) + a(cx + d)) - u(c(ax - b) + a(cx - d))}{v^2} \]

5. **Simplify the Expression**:
This step involves expanding and simplifying the numerator. You will collect like terms and factor where possible.

6. **Final Expression**:
After simplification, you will arrive at the final expression for \( \frac{dy}{dx} \).