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?

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.