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.