The function given is:

$y = x^4 e^x$

To find the derivative, we need to apply the product rule since the function is a product of two functions $x^4$ and $e^x$.

The product rule states:

$\frac{d}{dx}(uv) = u'v + uv'$

Here, $u = x^4$ and $v = e^x$.

1. The derivative of $u = x^4$ is $u' = 4x^3$.
2. The derivative of $v = e^x$ is $v' = e^x$.

Now, applying the product rule:

$\frac{dy}{dx} = u'v + uv' = (4x^3)(e^x) + (x^4)(e^x)$

Factoring out $e^x$:

$\frac{dy}{dx} = e^x(4x^3 + x^4)$

Thus, the derivative of the function is:

$\frac{dy}{dx} = e^x(x^4 + 4x^3)$

Would you like more details on the product rule or any other related concept?

Here are five questions that expand on this:

1. How does the product rule differ from the chain rule in calculus?
2. Why does $e^x$ have a unique property where its derivative remains $e^x$?
3. Can we simplify the expression further for different values of $x$?
4. How does the behavior of the derivative change for large values of $x$?
5. How can we apply the product rule to more complex functions, like trigonometric-exponential combinations?

Tip: When applying the product rule, always differentiate both functions carefully and ensure that you keep the original structure intact when combining the terms.