To find the derivative of the function $y = e^{x^2}$, we will use the chain rule:

Step-by-step process:

1. Write the function: $y = e^{x^2}$

2. Apply the chain rule: The derivative of $e^u$ with respect to $x$ is $e^u \cdot \frac{du}{dx}$, where $u = x^2$.

3. Differentiate $x^2$: The derivative of $x^2$ with respect to $x$ is $2x$.

4. Combine results: $\frac{dy}{dx} = e^{x^2} \cdot 2x$

Final Answer:

$\frac{dy}{dx} = 2x e^{x^2}$

Would you like a detailed explanation of the chain rule or help with a related concept? Here are five follow-up questions:

1. What is the derivative of $y = e^{x^3}$?
2. How do you apply the chain rule to more complex functions?
3. Can we find the second derivative of $y = e^{x^2}$?
4. What is the derivative of $y = e^{x^2 + 3x}$?
5. How does the chain rule relate to implicit differentiation?

Tip: Always look for nested functions in derivatives and use the chain rule step-by-step!