Find the derivative of :
`g(x)=x(x-3)(x^(2)+x)`.

Answer

To find the derivative of the function \( g(x) = x(x - 3)(x^2 + x) \), we will follow these steps:

### Step 1: Rewrite the function
Start by rewriting the function in a more manageable form. We can first expand the product of the first two factors:

\[ g(x) = x(x - 3)(x^2 + x) = x(x^2 - 3x)(x^2 + x) \]

### Step 2: Expand the function
Now, we will expand \( g(x) \):

1. Multiply \( x \) by \( (x - 3) \):
\[ x(x - 3) = x^2 - 3x \]

2. Now multiply this result by \( (x^2 + x) \):
\[ g(x) = (x^2 - 3x)(x^2 + x) \]

Using the distributive property (FOIL method):
\[ g(x) = x^2 \cdot x^2 + x^2 \cdot x - 3x \cdot x^2 - 3x \cdot x \]
\[ g(x) = x^4 + x^3 - 3x^3 - 3x^2 \]
\[ g(x) = x^4 - 2x^3 - 3x^2 \]

### Step 3: Differentiate the function
Now we differentiate \( g(x) \) using the power rule:

\[ g'(x) = \frac{d}{dx}(x^4) - \frac{d}{dx}(2x^3) - \frac{d}{dx}(3x^2) \]

Calculating each derivative:
\[ g'(x) = 4x^3 - 6x^2 - 6x \]

### Step 4: Final result
Thus, the derivative of the function \( g(x) \) is:

\[ g'(x) = 4x^3 - 6x^2 - 6x \]