To find the derivative of the function \( f(x) = (3x + 3)e^x \), we will use the product rule. The product rule states that if \( u(x) \) and \( v(x) \) are functions of \( x \), then the derivative \( (uv)' = u'v + uv' \).

Let:
- \( u = 3x + 3 \)
- \( v = e^x \)

First, we calculate the derivatives:
- \( u' = 3 \)
- \( v' = e^x \)

Now apply the product rule:
\[
f'(x) = u'v + uv' = (3)(e^x) + (3x + 3)(e^x)
\]

Factor out \( e^x \):
\[
f'(x) = e^x(3 + (3x + 3)) = e^x(3x + 6)
\]

Thus, the derivative is:
\[
f'(x) = e^x(3x + 6)
\]

Would you like more details on this process or have any specific questions? Here are some related questions you might find interesting:

1. What is the second derivative of the function?
2. How can we evaluate the derivative at a specific point, say \( x = 1 \)?
3. What is the behavior of \( f(x) \) as \( x \to \infty \)?
4. How does the product rule compare to the quotient rule?
5. Can you explain how to find critical points using the derivative? 

**Tip:** Always remember to check the conditions for applying the product rule or any other differentiation rules.

Learn how to differentiate the function f(x) = (3x + 3) * e^x by applying the product rule. This solution includes step-by-step differentiation, simplifying the expression, and factoring. Perfect for high school students studying calculus and differentiation techniques.

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