Infinite differentiability of a function with a removable discontinuity

Question

How would I prove that $\frac x{e^x-1}$ is infinitely differentiable? (This question came up since the No 1 answer in Maclaurin series for $\frac{x}{e^x-1}$ states that the function is infinitely differentiable, which - at least to me - isn't obvious at the first glance. Both the function and all its derivatives have a power of $e^x-1$ in the denominator which causes a (removable) discontinuity in $x=0$, but I suppose the answer can't be to look the removability of this discontinuity for all derivatives, right?)

Answer 1

Hint: Look at $g(x) = (e^x-1)/x$ instead. Then $g$ equals a power series that converges on all of $\mathbb R,$ hence is $C^\infty.$ Show that $g$ is never $0.$ Therefore …

Answer 2

Note that, if $f(x)$ is infinitely often differentiable and $f(0) \neq 0$ then $\frac{1}{f(x)}$ is infinitely often differentiable in $x=0$. This follows by induction, since you can show by induction that the $n$-th derivative of $\frac{1}{f(x)}$ is of the form $\frac{p(f, f^\prime, ..., f^{(n)})}{f^{2n}}$ with a polynomial $p$ of $n+1$ variables.

Apply this to $f(x)= \frac{e^x-1}{x}$