To find the derivative of the function $e^{-2}$, we need to apply the chain rule.

Given: $y = e^{-2}$

The exponent $-2$ is a constant, and the derivative of an exponential function $e^u$ with respect to $u$ is $e^u \cdot \frac{du}{dx}$.

In this case, $u = -2$, which is a constant. Therefore, $\frac{du}{dx} = 0$.

Applying the chain rule: $\frac{dy}{dx} = e^{-2} \cdot \frac{d(-2)}{dx}$

Since $\frac{d(-2)}{dx} = 0$: $\frac{dy}{dx} = e^{-2} \cdot 0 = 0$

So, the derivative of $e^{-2}$ is: $(e^{-2})' = 0$

Would you like more details or have any questions?

Here are 5 related questions:

1. What is the derivative of $e^{3x}$?
2. How do you find the derivative of a constant function?
3. What is the chain rule in differentiation?
4. Can you explain how to differentiate $e^{x^2}$?
5. How does the derivative of $e^x$ change with different exponents?

Tip: When differentiating exponential functions, always consider if the exponent is a function of $x$ or a constant. This will determine whether you need to use the chain rule or not.