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!

Understand how to differentiate the function y = e^2sinx by applying the chain rule. This detailed step-by-step guide walks through the key concepts of differentiation, including the chain rule and related derivatives. Perfect for students in Grades 11-12 or early college calculus courses.

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