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!