We need to differentiate:

$y = e^{x^2}$

Step 1: Use Chain Rule

The function can be rewritten as:

$y = e^u, \quad \text{where } u = x^2$

The derivative of $e^u$ is $e^u$, so applying the chain rule:

$\frac{dy}{dx} = e^{x^2} \cdot \frac{d}{dx} (x^2)$

Step 2: Differentiate the Exponent

Since $\frac{d}{dx} (x^2) = 2x$, we get:

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

Final Answer:

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

Would you like a step-by-step breakdown or an example related to this?

Related Questions:

1. How do you differentiate $e^{x^3}$?
2. What is the second derivative of $e^{x^2}$?
3. How does the chain rule work in differentiation?
4. How do you integrate $e^{x^2}$?
5. What is the derivative of $e^{\sin x}$?

Tip:

When differentiating exponentials like $e^{f(x)}$, always apply the chain rule by multiplying by $f'(x)$.