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.