To differentiate $y = e^{2\sin{x}}$, we apply the chain rule.

Step-by-step solution:

1. The function is $y = e^{2\sin{x}}$, so the outer function is $e^u$, where $u = 2\sin{x}$.

   The derivative of $e^u$ with respect to $u$ is: $\frac{d}{du}(e^u) = e^u$

2. Now differentiate $u = 2\sin{x}$ with respect to $x$. $\frac{d}{dx}(2\sin{x}) = 2\cos{x}$

3. Finally, by the chain rule: $\frac{dy}{dx} = e^{2\sin{x}} \cdot 2\cos{x}$

So, the derivative is: $\frac{dy}{dx} = 2e^{2\sin{x}} \cos{x}$

Would you like further clarification on any part?

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Here are 5 related questions to extend your understanding:

1. How would you differentiate $y = e^{3\sin{x}}$?
2. What is the second derivative of $y = e^{2\sin{x}}$?
3. Can you differentiate $y = \sin{(e^{2x})}$?
4. How does the chain rule work in a product of two functions?
5. How do you apply the chain rule to a logarithmic function, like $\ln(e^{2x})$?

Tip: Always identify the inner and outer functions when using the chain rule!