You are asking to compute the derivative of $e^{2x}$ with respect to $x$. To solve this, we can apply the chain rule for differentiation.

Let’s compute the derivative step by step:

$\frac{d}{dx} \left( e^{2x} \right)$

The chain rule tells us to differentiate the outer function (the exponential) and then multiply by the derivative of the inner function (the exponent). In this case:

1. The derivative of $e^{2x}$ is $e^{2x}$ (because the derivative of an exponential function is the function itself).
2. Now, multiply by the derivative of $2x$, which is 2.

So, the full derivative is:

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

Would you like more details, or do you have any further questions?

Here are 5 related questions:

1. What is the derivative of $e^{3x}$?
2. How do you apply the chain rule to $e^{x^2}$?
3. What is the second derivative of $e^{2x}$?
4. How do you compute the derivative of $e^{2x + 1}$?
5. How would you differentiate $e^{-x}$?

Tip: When differentiating exponential functions with non-trivial exponents, always remember to apply the chain rule correctly.