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.