Differentiate. \[ y=x^{4} e^{x} \] \( y^{\prime}=\square \)

To differentiate the function \( y = x^{4} e^{x} \), you can use the product rule, which states that if you have two functions multiplied together, \( u \cdot v \), the derivative is given by \( u'v + uv' \). Here, \( u = x^{4} \) and \( v = e^{x} \). The derivative of \( u \) is \( u' = 4x^{3} \), and the derivative of \( v \) is \( v' = e^{x} \). Applying the product rule: \[ y' = u'v + uv' = (4x^{3})(e^{x}) + (x^{4})(e^{x}) \] Factor out \( e^{x} \): \[ y' = e^{x}(4x^{3} + x^{4}) = e^{x}x^{3}(4 + x) \] So, the final answer is: \[ y' = e^{x} x^{3} (4 + x) \]