Show there is $r>0$ and continuously differentiable $f: (-r,r) \to \mathbb R$ with $f(0)=0$, such that the equation $$f(x)^2x+2x^2e^{f(x)}=f(x)$$ is fulfilled. Calculate $f'(0)$.
Right now I learned about the Implicit Function Theorem and I guess I need to make of it in this case. Yet I am having big issues on how to even approach this problem. Like the function being continuously differentiable is needed for the Theorem to be applied, so how can I prove that property? Do I have to state the function $f$ explicitly and prove the required properties after that one by one?