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.