https://file.aiquickdraw.com/mathbot/file/1727668640260a1qtuvcm.jpg

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.

This problem demonstrates how to differentiate the function y = 3x/e^x using the quotient rule. Key concepts include the quotient rule, derivatives of exponential functions, and simplifying the final result. Ideal for students in grades 11-12 or early college calculus.

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