`h(x)=(x+1)(x+2)(x+3)`. find `dy/dx`

Answer

To find the derivative \( \frac{dy}{dx} \) of the function \( h(x) = (x+1)(x+2)(x+3) \), we will follow these steps:

### Step 1: Expand the function
First, we need to expand the expression \( h(x) \).

\[ h(x) = (x+1)(x+2)(x+3) \]

We can start by multiplying the first two factors:

\[ (x+1)(x+2) = x^2 + 2x + x + 2 = x^2 + 3x + 2 \]

Now, we multiply this result by the third factor \( (x+3) \):

\[ h(x) = (x^2 + 3x + 2)(x + 3) \]

Now, we distribute \( (x + 3) \):

\[ = x^2(x + 3) + 3x(x + 3) + 2(x + 3) \]
\[ = x^3 + 3x^2 + 3x^2 + 9x + 2x + 6 \]
\[ = x^3 + 6x^2 + 11x + 6 \]

So, we have:

\[ h(x) = x^3 + 6x^2 + 11x + 6 \]

### Step 2: Differentiate the function
Now, we differentiate \( h(x) \) with respect to \( x \):

\[ \frac{dy}{dx} = \frac{d}{dx}(x^3 + 6x^2 + 11x + 6) \]

Using the power rule \( \frac{d}{dx}(x^n) = nx^{n-1} \):

\[ \frac{dy}{dx} = 3x^2 + 12x + 11 + 0 \]

### Final Answer
Thus, the derivative \( \frac{dy}{dx} \) is:

\[ \frac{dy}{dx} = 3x^2 + 12x + 11 \]

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