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The problem asks for the derivative of the function \( f(x) = \exp(x^2) \).

### Step-by-step differentiation:

1. **Function to differentiate:**
   \[
   f(x) = \exp(x^2)
   \]
   This is an exponential function where the exponent is a function of \(x\), specifically \( x^2 \).

2. **Chain Rule:**
   To differentiate this, we apply the chain rule. The chain rule states that if you have a composite function, \( f(g(x)) \), the derivative is:
   \[
   f'(x) = f'(g(x)) \cdot g'(x)
   \]

   In our case:
   - The outer function is \( \exp(u) \), where \( u = x^2 \).
   - The derivative of \( \exp(u) \) is \( \exp(u) \), so we have:
     \[
     \frac{d}{dx} \exp(x^2) = \exp(x^2) \cdot \frac{d}{dx}(x^2)
     \]
   - The derivative of \( x^2 \) is \( 2x \).

3. **Final derivative:**
   \[
   f'(x) = \exp(x^2) \cdot 2x
   \]

Thus, the correct derivative is:
\[
f'(x) = 2x \exp(x^2)
\]

### Correct answer from the options:

The correct choice is:
\[
f'(x) = 2x \exp(x^2)
\]

Do you want further details on the chain rule or have other questions?

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Here are five related questions:
1. How would you differentiate \( f(x) = \exp(3x^2) \)?
2. What is the derivative of \( f(x) = \exp(x^3) \)?
3. How does the chain rule apply when differentiating a nested exponential function?
4. What is the second derivative of \( f(x) = \exp(x^2) \)?
5. How do you differentiate a function of the form \( f(x) = e^{g(x)} \)?

**Tip:** Remember that the chain rule is key when differentiating composite functions like exponentials where the exponent is not just \( x \) but a more complex expression.

Hvad er den afledede af funktionen f(x) = exp(x^2)?

Learn how to differentiate the function f(x) = exp(x^2) using the chain rule. This problem covers key calculus concepts, including exponential functions and the application of the chain rule. Suitable for students in Grades 11-12 or equivalent.

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