Differentiate the following w.r.t. x or t or u as the case may be:
1. `(ax+ b) (cx+d)`

Answer

To differentiate the function \( y = (ax + b)(cx + d) \) with respect to \( x \), we will use the product rule of differentiation. The product rule states that if you have two functions \( u \) and \( v \), then the derivative of their product is given by:

\[ \frac{dy}{dx} = u \frac{dv}{dx} + v \frac{du}{dx} \]

### Step-by-step Solution:

1. **Identify the Functions**:
Let \( u = ax + b \) and \( v = cx + d \).

2. **Differentiate \( u \) and \( v \)**:
- The derivative of \( u \) with respect to \( x \) is:
\[ \frac{du}{dx} = a \]
- The derivative of \( v \) with respect to \( x \) is:
\[ \frac{dv}{dx} = c \]

3. **Apply the Product Rule**:
Now, substitute \( u \), \( v \), \( \frac{du}{dx} \), and \( \frac{dv}{dx} \) into the product rule formula:
\[ \frac{dy}{dx} = u \frac{dv}{dx} + v \frac{du}{dx} \]
Substituting the values:
\[ \frac{dy}{dx} = (ax + b)(c) + (cx + d)(a) \]

4. **Simplify the Expression**:
Now, we will expand the expression:
\[ \frac{dy}{dx} = c(ax + b) + a(cx + d) \]
Expanding both terms:
\[ = acx + bc + acx + ad \]
Combine like terms:
\[ = 2acx + (bc + ad) \]

5. **Final Result**:
The derivative of \( y = (ax + b)(cx + d) \) with respect to \( x \) is:
\[ \frac{dy}{dx} = 2acx + (bc + ad) \]