To differentiate the function \( y = e^{-x^2} \), we will use the chain rule.

### Solution:
Given \( y = e^{-x^2} \), let's define an inner function:
- \( u = -x^2 \)

Then, the derivative of \( e^u \) with respect to \( u \) is simply \( e^u \), and the derivative of \( u \) with respect to \( x \) is:
\[
\frac{du}{dx} = -2x
\]

Now, applying the chain rule:
\[
\frac{dy}{dx} = \frac{d}{du}(e^u) \cdot \frac{du}{dx} = e^{-x^2} \cdot (-2x)
\]

Thus, the derivative is:
\[
\frac{dy}{dx} = -2x e^{-x^2}
\]

Would you like more details on this process or have any questions?

### Related Questions:
1. How does the chain rule apply to more complex functions like \( e^{f(x)} \)?
2. What happens if the exponent was \( x^2 \) instead of \( -x^2 \)?
3. How do you differentiate other exponential functions like \( e^{x^3} \)?
4. How would the differentiation change if there were a constant multiplied by \( -x^2 \) in the exponent?
5. Can you apply the product rule with the chain rule for functions like \( x \cdot e^{-x^2} \)?

### Tip:
Whenever differentiating composite functions, identify the "inner" and "outer" functions to apply the chain rule correctly.

Learn how to differentiate the function y = e^(-x^2) using the chain rule in this step-by-step solution. This calculus problem involves exponential functions and highlights the importance of the chain rule in differentiation. Suitable for high school students in Grades 11-12.

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