In a previous question on this forum I described my issues on using the assumption of the solution being in the form of a Fourier series to solve the following differential equation.
$$ \frac{\mathrm{d}f}{\mathrm{d}x} = f(x) \qquad f(x) = \frac{A_0}{2} + \sum_{n=1}^\infty\left[A_0\cos\left(\frac{n\pi x}{L}\right) + B_0\sin\left(\frac{n\pi x}{L}\right)\right] $$
Upon term-by-term differentiation the assumed form $f(x)$ and substituting into the original equation a trivial solution yielded. This was explained to be that the term-by-term differentiation is essentially imposing a boundary condition that $f(L) = f(-L)$.
My question is that is there a way to find derivatives of Fourier series such that $f(-L) \ne f(L)$, I understand that the derivative of such a Fourier series will be undefined at $x = (2n+1)L$ where $n\in\mathbb{Z}$, but can an expression be defined for the derivative of the Fourier series in-between such values.