# How to guarantee if a real function is $C^\infty$ without using $f^n$

I know that function $$f(x)=x+\frac{1}{1+e^x}$$ is $$C^\infty$$ but I wanna prove that.

Question

Is there something that I could use to guarantee this without actually calculating $$f^{(n)}$$ ?

It is the sum of a trivially $$\mathcal C^\infty$$ function and the reciprocal of another $$\mathcal C^\infty$$ function which never vanishes. The rules of computation of the derivative of a reciprocal show this reciprocal is also $$\mathcal C^\infty$$.