To find the derivative of $y = e^{\sin(2x)}$, we'll use the chain rule.

Steps:

1. Outer Function: The outer function is $e^u$, where $u = \sin(2x)$. The derivative of $e^u$ with respect to $u$ is simply $e^u$.

   So, $\frac{d}{du} e^u = e^u$.

2. Inner Function: The inner function is $\sin(2x)$. We need to differentiate this with respect to $x$.

   The derivative of $\sin(2x)$ with respect to $x$ is $2\cos(2x)$, using the chain rule.

Thus, applying the chain rule:

$\frac{dy}{dx} = e^{\sin(2x)} \cdot 2\cos(2x)$

Final Answer:

$\frac{dy}{dx} = 2\cos(2x) \cdot e^{\sin(2x)}$

Let me know if you'd like more details or have any further questions!

Here are some related questions to explore:

1. How do we apply the chain rule in general?
2. What is the derivative of $y = e^{\cos(2x)}$?
3. How does the derivative change if the exponent is different, like $e^{\sin(3x)}$?
4. How do you find the derivative of composite functions that involve trigonometric functions?
5. Can the chain rule be used for higher-order derivatives?

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