Calculate the derivative of the following functions.
g(x) = x / e^3x

Verified step by step guidance

1. Step 1: Identify the function g(x) = \frac{x}{e^{3x}} as a quotient of two functions, where the numerator is f(x) = x and the denominator is h(x) = e^{3x}.
2. Step 2: Recall the Quotient Rule for derivatives, which states that if you have a function g(x) = \frac{f(x)}{h(x)}, then its derivative g'(x) is given by \frac{f'(x)h(x) - f(x)h'(x)}{(h(x))^2}.
3. Step 3: Calculate the derivative of the numerator, f'(x). Since f(x) = x, its derivative is f'(x) = 1.
4. Step 4: Calculate the derivative of the denominator, h'(x). Since h(x) = e^{3x}, use the chain rule to find h'(x) = 3e^{3x}.
5. Step 5: Substitute f(x), f'(x), h(x), and h'(x) into the Quotient Rule formula to find g'(x) = \frac{1 \cdot e^{3x} - x \cdot 3e^{3x}}{(e^{3x})^2}.