To differentiate $y = e^{-2x}$, we apply the chain rule.

The chain rule states that if $y = e^{u(x)}$, then $\frac{dy}{dx} = e^{u(x)} \cdot \frac{du}{dx}$, where $u(x)$ is a function of $x$.

Here, $u(x) = -2x$. Now, we differentiate $u(x)$ with respect to $x$:

$\frac{du}{dx} = -2$

Now, applying the chain rule:

$\frac{dy}{dx} = e^{-2x} \cdot (-2) = -2e^{-2x}$

So, the derivative of $y = e^{-2x}$ is:

$\frac{dy}{dx} = -2e^{-2x}$

Let me know if you'd like more details or have any questions! Here are 5 related questions you can explore:

1. What is the derivative of $y = e^{3x}$?
2. How does the chain rule work when differentiating functions like $y = \sin(2x)$?
3. What is the derivative of $y = e^{x^2}$?
4. How do you differentiate a function like $y = 5e^{x}$?
5. What is the second derivative of $y = e^{-2x}$?

Tip: When using the chain rule, always differentiate the outer function first and then multiply by the derivative of the inner function.