Sorry to contraddict, but the Laplace transform of a derivative is not that simple... remember that we are transforming between time and complex frequency, so the "memory" included in the derivative is to be found explicitly in the Laplace domain. Otherwise you are not gonna be able to solve the Cauchy problem deriving form your differential equation:
\(L(f'(t)) = s F(s) + f(t)|_{s=0^+}\)
so the correct solution of the problem (given you don't know any initial states) is: