The derivative of \( e^{-t} \) with respect to \( t \) can be computed using the chain rule. Here's the solution:

\[
\frac{d}{dt} \left( e^{-t} \right) = e^{-t} \cdot \frac{d}{dt} (-t) = -e^{-t}
\]

So, the derivative of \( e^{-t} \) is:

\[
-\ e^{-t}
\]

Would you like further details or explanations on how the chain rule applies here?

### Related Questions:
1. What is the derivative of \( e^{at} \) where \( a \) is a constant?
2. How do you find the second derivative of \( e^{-t} \)?
3. What is the derivative of \( e^{-t^2} \)?
4. How is the chain rule used to differentiate more complex functions involving exponentials?
5. How can derivatives of exponentials be applied in physics or economics?

**Tip:** The chain rule is essential when differentiating composite functions—always remember to multiply by the derivative of the inner function.

Explore how to compute the derivative of e^-t using the chain rule. This calculus problem demonstrates how the chain rule applies to differentiating exponential functions, making it suitable for high school students in Grades 11-12.

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