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:

 B_{n} is the  \sin{\frac{n\pi x}{L}}
term of f'(x),

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