Differentiate the function. \[ f(x)=x^{3}(x+4) \] \( f^{\prime}(x)=\square \)

Let \( f(x) = x^3(x+4) \). 1. Identify the two functions for the product rule: Let \( u(x) = x^3 \) and \( v(x) = x+4 \). 2. Compute the derivatives: \[ u'(x) = 3x^2 \quad \text{and} \quad v'(x) = 1 \] 3. Apply the product rule: The product rule states that \[ f'(x) = u'(x)v(x) + u(x)v'(x) \] Thus, \[ f'(x) = 3x^2 (x+4) + x^3(1) \] 4. Simplify the expression: \[ 3x^2(x+4) + x^3 = 3x^3 + 12x^2 + x^3 = 4x^3 + 12x^2 \] 5. Factor the simplified expression: Factor out \( 4x^2 \) from the terms: \[ 4x^3 + 12x^2 = 4x^2\left(x + 3\right) \] Therefore, the derivative of the function is: \[ f'(x)=4x^2\left(x+3\right) \]