Differentiate the following w.r.t. x :
`(x+3)^(2)(x+4)^(3)(x+5)^(4)`

Answer

To differentiate the function \( y = (x+3)^2 (x+4)^3 (x+5)^4 \) with respect to \( x \), we can use the product rule and the chain rule. Here’s a step-by-step solution:

### Step 1: Identify the function
We have the function:
\[ y = (x+3)^2 (x+4)^3 (x+5)^4 \]

### Step 2: Apply the product rule
Since \( y \) is a product of three functions, we will use the product rule. The product rule states that if you have a product of functions \( u, v, w \), then:
\[ \frac{d}{dx}(uvw) = u'vw + uv'w + uvw' \]
Let:
- \( u = (x+3)^2 \)
- \( v = (x+4)^3 \)
- \( w = (x+5)^4 \)

### Step 3: Differentiate each function
Now we need to differentiate each function:
1. \( u' = \frac{d}{dx}[(x+3)^2] = 2(x+3) \)
2. \( v' = \frac{d}{dx}[(x+4)^3] = 3(x+4)^2 \)
3. \( w' = \frac{d}{dx}[(x+5)^4] = 4(x+5)^3 \)

### Step 4: Substitute into the product rule
Now we substitute \( u, v, w, u', v', w' \) into the product rule:
\[ \frac{dy}{dx} = u'vw + uv'w + uvw' \]
Substituting the values:
\[ \frac{dy}{dx} = 2(x+3)(x+4)^3(x+5)^4 + (x+3)^2 \cdot 3(x+4)^2(x+5)^4 + (x+3)^2(x+4)^3 \cdot 4(x+5)^3 \]

### Step 5: Simplify the expression
Now we can factor out the common terms:
\[ \frac{dy}{dx} = (x+3)^2 (x+4)^2 (x+5)^3 \left[ 2(x+4)(x+5) + 3(x+3)(x+5) + 4(x+3)(x+4) \right] \]

### Step 6: Expand the terms inside the brackets
Now we expand the terms inside the brackets:
1. \( 2(x+4)(x+5) = 2(x^2 + 9x + 20) = 2x^2 + 18x + 40 \)
2. \( 3(x+3)(x+5) = 3(x^2 + 8x + 15) = 3x^2 + 24x + 45 \)
3. \( 4(x+3)(x+4) = 4(x^2 + 7x + 12) = 4x^2 + 28x + 48 \)

Adding these together:
\[ (2x^2 + 18x + 40) + (3x^2 + 24x + 45) + (4x^2 + 28x + 48) = 9x^2 + 70x + 133 \]

### Final Result
Thus, the derivative is:
\[ \frac{dy}{dx} = (x+3)^2 (x+4)^2 (x+5)^3 (9x^2 + 70x + 133) \]