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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 $f^{(n)}$ ?

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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 $\mathcal C^\infty$.

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