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The given function is \( y = 3x e^x \). To find the derivative, we will apply the **product rule**, which states:

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

Here, \( u = 3x \) and \( v = e^x \). Let's compute the derivative:

1. The derivative of \( u = 3x \) is \( u' = 3 \).
2. The derivative of \( v = e^x \) is \( v' = e^x \).

Now, applying the product rule:

\[
y' = (3x)' e^x + 3x (e^x)'
\]
\[
y' = 3 e^x + 3x e^x
\]

Thus, the derivative of the function is:

\[
y' = 3 e^x + 3x e^x
\]

Would you like further details or have any questions? Here are some related questions to explore:

1. What is the product rule in differentiation?
2. How do you differentiate exponential functions like \( e^x \)?
3. Can you explain how to find the second derivative of \( y = 3x e^x \)?
4. What is the significance of the exponential function in calculus?
5. How would the derivative change if the function were \( y = x^2 e^x \)?

**Tip:** When using the product rule, remember that each term must be differentiated separately, and then combined according to the rule.

Find the derivative of the function y = 3x e^x.

Learn how to find the derivative of the function y = 3x e^x using the product rule in calculus. This problem explains key differentiation concepts, including the product rule and how to handle exponential functions. Suitable for high school and early college students.

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