In the lectures, lecturer just said,

f(x) needs to be continuous, 2L periodic, piecewise continuously differentiable on (-L,L)

is this the condition for differentiability of Fourier Series:

$\displaystyle B_{n} $ is the $\displaystyle \sin{\frac{n\pi x}{L}} $

term of f'(x),

then, $\displaystyle B_{n}=-\frac{n\pi a_{n}}{L} $