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The given function is:

\[
y = \frac{e^{2x}}{e^x + 3}
\]

This is a rational function involving exponential terms. How would you like to proceed with it? For instance, we could:

1. Find the first derivative (\(y'\)) using differentiation.
2. Analyze the behavior of the function, such as finding asymptotes or critical points.
3. Determine the limits as \(x \to \infty\) or \(x \to -\infty\).
4. Find the second derivative to analyze concavity.
5. Solve for \(x\) when \(y = 0\) or another specific value.

Would you like details on any of these, or do you have a different request for this function?

Also, here are some questions that could help guide further exploration:
1. What is the derivative of \( y = \frac{e^{2x}}{e^x + 3} \)?
2. What are the critical points of this function?
3. What is the behavior of the function as \( x \to \infty \)?
4. Can we simplify the given function in any way?
5. What is the second derivative of the function?

**Tip**: When differentiating a quotient, always apply the quotient rule: \( \frac{f}{g} \) becomes \( \frac{f'g - fg'}{g^2} \).

Explore the differentiation of y = e^(2x) / (e^x + 3) using the quotient rule, a key calculus technique. This example includes exponential functions and discusses critical points, limits, and function behavior. Ideal for high school calculus students in Grades 11-12.

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