UPDATE: I do not want to use the Derivative or D functions.
You must not, you can use Cauchy's integral to differentiate, see Complex variable methods in Numerical Differentiation, you can use the ND implementation in the Mathematica package NumericalCalculus` ...
ND
You do know that Mathematica can compute derivatives:
f'[x]=D[f[x],x] and f''[x]=D[f[x],{x,2}]
