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To find the derivative of the given function:

\[
g(x) = \frac{e^x}{3 - 3x}
\]

we will use the **quotient rule**, which states that if \( g(x) = \frac{f(x)}{h(x)} \), then:

\[
g'(x) = \frac{f'(x)h(x) - f(x)h'(x)}{(h(x))^2}
\]

### Step-by-step solution:
1. Define \( f(x) = e^x \) and \( h(x) = 3 - 3x \).
2. Compute the derivatives:
   - \( f'(x) = \frac{d}{dx}[e^x] = e^x \)
   - \( h'(x) = \frac{d}{dx}[3 - 3x] = -3 \)

3. Apply the quotient rule:

\[
g'(x) = \frac{e^x(3 - 3x) - e^x(-3)}{(3 - 3x)^2}
\]

4. Simplify the numerator:

\[
g'(x) = \frac{e^x(3 - 3x) + e^x(3)}{(3 - 3x)^2} = \frac{e^x(3 - 3x + 3)}{(3 - 3x)^2} = \frac{e^x(6 - 3x)}{(3 - 3x)^2}
\]

Thus, the derivative is:

\[
g'(x) = \frac{e^x(6 - 3x)}{(3 - 3x)^2}
\]

Would you like more details or explanations on any step?

Here are 5 related questions you might find interesting:
1. How do we handle higher-order derivatives of \( g(x) \)?
2. What is the behavior of \( g(x) \) as \( x \) approaches infinity?
3. Can we simplify \( g(x) \) further using algebraic techniques before taking the derivative?
4. How can we apply the product rule to derivatives involving two functions?
5. What are the real-world applications of functions that involve exponentials and rational expressions?

**Tip**: Always simplify the expression as much as possible before applying differentiation rules, especially for complex functions.

Find the derivative of the function g(x) = e^x / (3 - 3x).

Learn how to find the derivative of the function g(x) = e^x / (3 - 3x) using the quotient rule. This problem covers key calculus concepts such as differentiation, exponential functions, and the quotient rule. Suitable for college-level calculus students.

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