To find the second derivative \( y'' \) of the function \( y = e^{2x} \), we can proceed as follows:

1. The first derivative of \( y = e^{2x} \) with respect to \( x \) is calculated using the chain rule:
   \[
   y' = \frac{d}{dx} \left( e^{2x} \right) = 2e^{2x}
   \]
2. The second derivative \( y'' \) is simply the derivative of \( y' \):
   \[
   y'' = \frac{d}{dx} \left( 2e^{2x} \right) = 4e^{2x}
   \]

Thus, the second derivative is:
\[
y'' = 4e^{2x}
\]

Would you like more details or have any questions about this? 

Here are five related questions to expand your understanding:

1. What is the third derivative of \( y = e^{2x} \)?
2. How does the chain rule apply to derivatives of exponential functions?
3. What is the derivative of \( e^{ax + b} \), where \( a \) and \( b \) are constants?
4. Can you explain why the exponential function \( e^{2x} \) keeps its form after differentiation?
5. How would you compute higher-order derivatives of \( e^{nx} \)?

**Tip:** The exponential function is unique in calculus because it always retains its exponential form when differentiated.

Explore how to find the second derivative of e^2x using the chain rule. This detailed guide walks through the process, including key differentiation formulas and theorems. Suitable for high school students in Grades 10-12.

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