Differentiate the following by :
`(x+1)(x+2)(x+3)`

Answer

To differentiate the expression \((x+1)(x+2)(x+3)\), we will follow these steps:

### Step 1: Expand the expression
First, we will expand the product \((x+1)(x+2)(x+3)\).

1. Multiply the first two factors:
\[ (x+1)(x+2) = x^2 + 2x + x + 2 = x^2 + 3x + 2 \]

2. Now multiply this result with the third factor \((x+3)\):
\[ (x^2 + 3x + 2)(x + 3) \]
Using the distributive property:
\[ = x^2(x + 3) + 3x(x + 3) + 2(x + 3) \]
\[ = x^3 + 3x^2 + 3x^2 + 9x + 2x + 6 \]
\[ = x^3 + 6x^2 + 11x + 6 \]

### Step 2: Differentiate the expanded expression
Now, we differentiate the polynomial \(y = x^3 + 6x^2 + 11x + 6\).

Using the power rule of differentiation, which states that \(\frac{d}{dx}(x^n) = nx^{n-1}\):

1. Differentiate \(x^3\):
\[ \frac{d}{dx}(x^3) = 3x^2 \]

2. Differentiate \(6x^2\):
\[ \frac{d}{dx}(6x^2) = 6 \cdot 2x^{2-1} = 12x \]

3. Differentiate \(11x\):
\[ \frac{d}{dx}(11x) = 11 \]

4. Differentiate the constant \(6\):
\[ \frac{d}{dx}(6) = 0 \]

Now, combine all these results:
\[ \frac{dy}{dx} = 3x^2 + 12x + 11 \]

### Final Result:
Thus, the derivative of \((x+1)(x+2)(x+3)\) is:
\[ \frac{dy}{dx} = 3x^2 + 12x + 11 \]

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