1. Problemet er at differentiere funktionen $$f(x) = \ln(x^2 + 1) + 2x$$. 2. Vi bruger reglen for differentiering af en sum: $$\frac{d}{dx}[u + v] = \frac{du}{dx} + \frac{dv}{dx}$$. 3. Differentieringsreglen for den naturlige logaritme er: $$\frac{d}{dx}[\ln(g(x))] = \frac{g'(x)}{g(x)}$$. 4. Differentier funktionen trin for trin: - For $$\ln(x^2 + 1)$$, sæt $$g(x) = x^2 + 1$$. - Differentier $$g(x)$$: $$g'(x) = 2x$$. - Anvend logaritmereglen: $$\frac{d}{dx}[\ln(x^2 + 1)] = \frac{2x}{x^2 + 1}$$. - Differentier $$2x$$: $$\frac{d}{dx}[2x] = 2$$. 5. Saml resultaterne: $$f'(x) = \frac{2x}{x^2 + 1} + 2$$. 6. Det er den afledte funktion for $$f(x)$$. Svar: $$\boxed{f'(x) = \frac{2x}{x^2 + 1} + 2}$$.