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

To differentiate the function \( f(x) = x^{3}(x + 4) \), you can use the product rule, which states that if you have a function defined as the product of two functions, say \( u(x) \) and \( v(x) \), then the derivative is given by \( u'v + uv' \). Here, let \( u = x^{3} \) and \( v = (x + 4) \). First, calculate the derivatives: - \( u' = 3x^{2} \) - \( v' = 1 \) Applying the product rule: \[ f'(x) = u'v + uv' = (3x^{2})(x + 4) + (x^{3})(1) \] Now expand and simplify the expression: \[ = 3x^{2}(x + 4) + x^{3} = 3x^{3} + 12x^{2} + x^{3} = 4x^{3} + 12x^{2} \] Thus, the derivative is: \[ f'(x) = 4x^{3} + 12x^{2} \] Now you have it, \( f^{\prime}(x) = 4x^{3} + 12x^{2} \)!