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The problem asks to find the second derivative of \( y = e^{-x} \).

Let's work through it step by step:

### Step 1: First Derivative

Given:
\[
y = e^{-x}
\]
Differentiate \( y \) with respect to \( x \) using the chain rule:

\[
y' = \frac{d}{dx}(e^{-x}) = e^{-x} \cdot (-1) = -e^{-x}
\]

### Step 2: Second Derivative

Now, differentiate \( y' = -e^{-x} \) again with respect to \( x \):

\[
y'' = \frac{d}{dx}(-e^{-x}) = -e^{-x} \cdot (-1) = e^{-x}
\]

### Final Answer:

\[
y'' = e^{-x}
\]

Let me know if you'd like any more details or have further questions.

### Related Questions:
1. What is the third derivative of \( y = e^{-x} \)?
2. How does the chain rule apply when differentiating exponential functions?
3. What is the general form of the \( n \)-th derivative of \( e^{-x} \)?
4. How does the derivative of \( e^{-x} \) compare to that of \( e^{x} \)?
5. What happens if we differentiate \( e^{kx} \), where \( k \) is a constant?

### Tip:
When differentiating exponentials of the form \( e^{kx} \), always apply the chain rule, multiplying by the derivative of the exponent.

This page provides a detailed step-by-step solution to finding the second derivative of y = e^(-x). Learn about the chain rule and how it applies to differentiating exponential functions. Suitable for advanced high school or college-level calculus students.

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