The image shows the function $y = \frac{3x}{e^x}$, and the task is to differentiate it.

To differentiate the function $y = \frac{3x}{e^x}$, we will apply the quotient rule, which is:

$\left( \frac{f(x)}{g(x)} \right)' = \frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2}$

Here, $f(x) = 3x$ and $g(x) = e^x$.

1. Differentiate $f(x) = 3x$: $f'(x) = 3$

2. Differentiate $g(x) = e^x$: $g'(x) = e^x$

Now, applying the quotient rule:

$y' = \frac{(3)(e^x) - (3x)(e^x)}{(e^x)^2}$

Simplify the numerator:

$y' = \frac{3e^x - 3xe^x}{e^{2x}}$

Factor out $3e^x$ from the numerator:

$y' = \frac{3e^x(1 - x)}{e^{2x}}$

Finally, simplify the expression:

$y' = \frac{3(1 - x)}{e^x}$

Thus, the derivative is:

$y' = \frac{3(1 - x)}{e^x}$

Would you like more details on any of these steps?

Here are five related questions to explore:

1. How does the quotient rule apply in other cases?
2. What is the significance of the exponential function $e^x$ in calculus?
3. Can the product rule be used in this scenario, and how would it compare?
4. How can we differentiate other types of rational functions?
5. What is the derivative of $e^x$ with other constants?

Tip: Always check if the function is best suited for the quotient rule or if it can be simplified first.