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The derivative of the given function \( \frac{dx}{dt} = \frac{2 \cos(2t) - \sin(2t)}{e^t} \) is:

\[
\frac{dx}{dt} = \left( -4 \sin(2t) - 2 \cos(2t) \right) e^{-t} - \left( - \sin(2t) + 2 \cos(2t) \right) e^{-t}
\]

Would you like further simplification or explanation on any specific part?

Here are five questions related to this topic:
1. How do we handle the product rule when differentiating more complex functions?
2. What is the role of the chain rule in differentiating trigonometric functions like \( \cos(2t) \)?
3. Why do we multiply by \( e^{-t} \) when differentiating a quotient involving exponentials?
4. How can this type of differentiation be applied in physics for modeling periodic phenomena?
5. How would you integrate the result obtained?

**Tip**: When differentiating quotients, applying the product rule with the negative exponent often simplifies the process!

Differentiate \( \frac{dx}{dt} = \frac{2 \cos(2t) - \sin(2t)}{e^t} \)

Learn how to differentiate the function \( \frac{2 \cos(2t) - \sin(2t)}{e^t} \) using the quotient and product rules. Explore the application of the chain rule in differentiating trigonometric expressions. This example is suitable for university-level calculus students.

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