The purpose of this paper is to establish some theorems on the representation of analytic functions by exponential derived bases (EDBs) and exponential integral bases (EIBs) in Fréchet spaces. Theorems are proved to show such that the representation is possible in closed disks, open disks, open regions surrounding closed disks, at the origin and for all entire functions. We studied the effectiveness property of exponential derived bases (EDBs) and exponential integral bases (EIBs). Also some results concerning the growth order and type of exponential derived bases (EDBs) and exponential integral bases (EIBs) are determined. Moreover the T_ρ property of EDBs is discussed in closed disk in Fréchet spaces. We provided this paper with examples, if the condition of the theorem is not achieved. Finally, some applications to the exponential derived bases (EDBs) and exponential integral bases (EIBs) of Bernoulli, Euler, Bessel, and Chebyshev polynomials have been studied in closed disk of in Fréchet spaces.