analysis - For which $alpha$ does this function have a positive solution?

For which $\alpha \in \mathbb{R}$ does $$e^{\alpha x}-1=x$$ have a positive solution
Hint; Consider Derivatives at $0$.

My attempt;
Now firstly I will rewrite this to form the function $$f(x)=e^{\alpha x}-1-x$$ Now we want to see for what $\alpha \in \mathbb{R}$ does $f(x)=0$ yield a positive solution.
Taking derivatives we get $$f'(x)=\alpha e^{\alpha x}-1$$ and a $x=0$ we get $$f'(0)=\alpha -1$$ .
Now I seem confused on how to proceed, any help would be appreciated.