1. The problem is to find the derivative of the function $$f(x) = e^{x^2}$$. 2. We use the chain rule for differentiation, which states that if $$f(x) = e^{g(x)}$$, then $$f'(x) = e^{g(x)} \cdot g'(x)$$. 3. Here, $$g(x) = x^2$$, so we find its derivative: $$g'(x) = 2x$$. 4. Applying the chain rule, we get: $$f'(x) = e^{x^2} \cdot 2x = 2x e^{x^2}$$. 5. Therefore, the correct derivative is $$f'(x) = 2x e^{x^2}$$. 6. Comparing with the options, the correct answer is d. $$f'(x) = 2x e^{x^2}$$.