1. Problem: Differentiate the function $$f(x) = \ln(x^2 + 1) + 2x$$. 2. Formula: To differentiate a sum, differentiate each term separately. For $$\ln(u)$$, the derivative is $$\frac{u'}{u}$$ where $$u$$ is a function of $$x$$. 3. Differentiate the first term: Let $$u = x^2 + 1$$. Then $$u' = 2x$$. So, $$\frac{d}{dx} \ln(x^2 + 1) = \frac{2x}{x^2 + 1}$$. 4. Differentiate the second term: $$\frac{d}{dx} 2x = 2$$. 5. Combine the derivatives: $$f'(x) = \frac{2x}{x^2 + 1} + 2$$. 6. Final answer: $$\boxed{f'(x) = \frac{2x}{x^2 + 1} + 2}$$