The derivative of \( e^{2x} \) with respect to \( x \) can be calculated using the chain rule.

Given:

\[
f(x) = e^{2x}
\]

1. The outer function is \( e^u \) where \( u = 2x \).
2. The derivative of \( e^u \) with respect to \( u \) is \( e^u \).
3. The derivative of \( u = 2x \) with respect to \( x \) is \( 2 \).

Using the chain rule:

\[
\frac{d}{dx} \left( e^{2x} \right) = e^{2x} \cdot \frac{d}{dx}(2x) = e^{2x} \cdot 2
\]

Thus, the derivative is:

\[
\frac{d}{dx} \left( e^{2x} \right) = 2e^{2x}
\]

Would you like more details or have any other questions?

Here are 5 related questions for further exploration:

1. What is the derivative of \( e^{3x} \)?
2. How do you apply the chain rule to more complex functions?
3. What is the second derivative of \( e^{2x} \)?
4. How do you differentiate \( e^{x^2} \)?
5. What is the derivative of \( e^{x} \) multiplied by another function?

**Tip**: Always apply the chain rule when differentiating compositions of functions!

Learn how to find the derivative of e^2x using the chain rule in calculus. This detailed explanation walks you through each step, including the derivative of exponential functions and the application of the chain rule. Suitable for students in Grades 11-12 or introductory college calculus.

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