To find \( f'(x) \) for the function \( f(x) = \sin(e^{2x}) \), we will use the chain rule. 

The chain rule states:
\[
\frac{d}{dx} [\sin(g(x))] = \cos(g(x)) \cdot g'(x)
\]

Here, \( g(x) = e^{2x} \), so we need to find the derivative of \( e^{2x} \) first:
\[
\frac{d}{dx} [e^{2x}] = 2e^{2x}
\]

Now, applying the chain rule:
\[
f'(x) = \cos(e^{2x}) \cdot 2e^{2x}
\]

Thus, the derivative of \( f(x) = \sin(e^{2x}) \) is:
\[
f'(x) = 2e^{2x} \cos(e^{2x})
\]

Let me know if you have any questions! Here are 5 related questions to further your understanding:

1. What is the derivative of \( f(x) = \cos(e^{2x}) \)?
2. How would the derivative change if \( f(x) = \sin(e^{3x}) \)?
3. Can you find the second derivative \( f''(x) \) for \( f(x) = \sin(e^{2x}) \)?
4. What is the general rule for differentiating composite functions?
5. How would you differentiate \( f(x) = \sin(e^{2x^2}) \)?

**Tip:** When applying the chain rule, always start by identifying the outermost function and work inward.

Learn how to find the derivative of the function f(x) = sin(e^2x) using the chain rule. This solution covers differentiation of trigonometric and exponential functions. Suitable for advanced high school students in Grades 11-12.

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