How do i prove f(0) = 0 (Calculus problem)

Question

I need some help to get started with this question.

$$f(x) + \ln(1+f(x)) = \frac{\sin x}{1+x}$$

Show $f(0) = 0$ and use that to find $f'(0)$.

Where do i start to show $f(0) = 0$? $f(x)$ inside $\ln(1+f(x))$ is confusing me a a lot.

Answer 1

For $x=0$, you get $$f(0)+\ln(1+f(0))=0$$

so $f(0)$ is solution of the equation $y+\ln(1+y)=0$. Now, the function $g : y \mapsto y +\ln(1+y)$ is strictly increasing on its domain, so it is injective. Moreover, $g(0)=0$, so $0$ is the only solution to $g(0)=0$. So $$f(0)=0$$