  This paper considers the $ p $ ($ p=1,2,3 $) order numerical differentiation
on function $ y $ in $ (0,2\pi) $. They are transformed into corresponding
Fredholm integral equation of the first kind. Computational schemes with
analytic solution formulas are designed using Galerkin method on trigonometric
basis. Convergence and divergence are all analysed in Corollaries 5.1, 5.2, and
a-priori error estimate is uniformly obtained in Theorem 6.1, 7.1, 7.2.
Therefore, the algorithm achieves the optimal convergence rate $ O(
\delta^{\frac{2\mu}{2\mu+1}} ) \ (\mu = \frac{1}{2} \ \textrm{or} \ 1)$ with
periodic Sobolev source condition of order $ 2\mu p $. Besides, we indicate a
noise-independent a-priori parameter choice when the function $ y $ possesses
the form of \begin{equation*}
  \sum^{p-1}_{k=0} a_k t^k + \sum^{N_1}_{k=1} b_k \cos k t + \sum^{N_2}_{k=1}
c_k \sin k t, \ b_{N_1}, c_{N_2} \neq 0, \end{equation*} In particular, in
numerical differentiations for functions above, good filtering effect (error
approaches 0) is displayed with corresponding parameter choice. In addition,
several numerical examples are given to show that even derivatives with
discontinuity can be recovered well.
