My understanding is that to get f'(0), you can't use the formula for f'(x) obtained by using the quotient rule, etc., because this only applies when . Instead, you need to look at right- and left-derivatives, defined by
respectively, at the point x=a=0. If both exist and are equal, then that is the value of the derivative at that point. I worked some of it on paper and had to use L'Hopital's rule, then got lazy and switched to Mathematica, anyway I found that f'(0)=0, but f''(0)=1/3, and I didn't try to find out the higher order derivatives.
Hope that helps.
EDIT: It seems that f(x) is infinitely differentiable, which makes consideration of right- and left-derivatives unnecessary. So to get that f''(0)=1/3, it is enough to evaluate .
EDIT 2: Did the problem require you to characterize all terms of the Maclaurin series, or just some of the first terms? Because even if I'm completely lazy and let Mathematica do all the work for me, I still don't see a pattern to characterize the terms...