The efficient computation of molecular integrals and their derivatives is a crucial step in molecular property evaluation in modern quantum chemistry. As an integral tensor decomposition technique, the density-fitting (DF) approach becomes a popular tool to reduce the memory and disk requirements for the electron repulsion integrals. In this study, an application programming interface (API) framework, denoted (MFW), for the computation of molecular integrals and their first derivatives, over contracted Gaussian functions, for the density-fitted methods is reported. The MFW is free software and it includes overlap, dipole, kinetic, potential, metric, and 3-index integrals, and their first derivatives. Furthermore, the MFW provides a smooth approach to build the Fock matrix and evaluate analytic gradients for the density-fitted methods. The MFW is a C++/Fortran hybrid code, which can take advantage of shared-memory parallel programming techniques. Our results demonstrate that the MFW is an efficient and user-friendly API for the computation of molecular integrals and their first derivatives.