(iii) Let `y= (x+1) (2x +3)^(3) (5x + 7)^(2)`

Answer

To differentiate the function \( y = (x + 1)(2x + 3)^3(5x + 7)^2 \), we will use the product rule and the chain rule of differentiation. Here’s a step-by-step solution:

### Step 1: Identify the components of the function
We have three factors in the function:
1. \( u = (x + 1) \)
2. \( v = (2x + 3)^3 \)
3. \( w = (5x + 7)^2 \)

### Step 2: Differentiate each component
1. Differentiate \( u \):
\[ u' = \frac{d}{dx}(x + 1) = 1 \]

2. Differentiate \( v \) using the chain rule:
\[ v' = \frac{d}{dx}((2x + 3)^3) = 3(2x + 3)^2 \cdot (2) = 6(2x + 3)^2 \]

3. Differentiate \( w \) using the chain rule:
\[ w' = \frac{d}{dx}((5x + 7)^2) = 2(5x + 7) \cdot (5) = 10(5x + 7) \]

### Step 3: Apply the product rule
The product rule states that if \( y = u \cdot v \cdot w \), then:
\[ y' = u'vw + uv'w + uvw' \]

Substituting the derivatives we found:
\[ y' = (1)(2x + 3)^3(5x + 7)^2 + (x + 1)(6(2x + 3)^2)(5x + 7)^2 + (x + 1)(2x + 3)^3(10(5x + 7)) \]

### Step 4: Simplify the expression
Now we can simplify the expression:
1. The first term is:
\[ (2x + 3)^3(5x + 7)^2 \]

2. The second term is:
\[ 6(x + 1)(2x + 3)^2(5x + 7)^2 \]

3. The third term is:
\[ 10(x + 1)(2x + 3)^3(5x + 7) \]

Combining these gives:
\[ y' = (2x + 3)^3(5x + 7)^2 + 6(x + 1)(2x + 3)^2(5x + 7)^2 + 10(x + 1)(2x + 3)^3(5x + 7) \]

### Final Answer
Thus, the derivative of the function \( y \) is:
\[ y' = (2x + 3)^3(5x + 7)^2 + 6(x + 1)(2x + 3)^2(5x + 7)^2 + 10(x + 1)(2x + 3)^3(5x + 7) \]