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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.

Find the derivative of the function y = x^4 * e^x.

Learn how to find the derivative of the function y = x^4 * e^x using the product rule. This solution covers essential calculus concepts, including differentiation and the unique properties of exponential functions. Suitable for students in Grades 11-12.

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