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.