"Bad" Fourier Series derivation

You have to think in terms of distributional derivatives, because in this sense you can switch series and derivative. Think $f$ as defined on the 1-torus. Then the distributional derivative of $f$ it is not $f$ itself, but $f-(e^\pi-e^{-\pi})\delta_{[\pi]}$, where $\delta_{[\pi]}$ is the delta Dirac in the point $[\pi]$ of the 1-torus. So, in distributional sense: $$\sum_{n=-\infty}^\infty inc_ne_n = f'=f-(e^\pi-e^{-\pi})\delta_{[\pi]}=\sum_{n=-\infty}^\infty c_ne_n - (e^\pi-e^{-\pi})\delta_{[\pi]}$$ i.e.: $$\sum_{n=-\infty}^\infty (in-1)c_ne_n=-(e^\pi-e^{-\pi})\delta_{[\pi]}=-(e^\pi-e^{-\pi})\sum_{n=-\infty}^\infty e^{in\pi}e_n \\ = -(e^\pi-e^{-\pi})\sum_{n=-\infty}^\infty \frac{(-1)^n}{2\pi}e_n.$$ Then: $$\forall n\in \mathbb{Z}, (in-1)c_n=-\frac{(-1)^{n}}{2\pi}(e^\pi-e^{-\pi})$$ i.e.: $$\forall n\in\mathbb{Z}, c_n=\frac{(-1)^{n}(e^\pi-e^{-\pi})}{2\pi(1-in)}.$$