This study introduces a new mixed fractional derivative that encompasses various types of fractional derivatives, including the Riemann–Liouville and Caputo derivatives with singular kernels, and the Caputo–Fabrizio, Atangana–Baleanu, and generalized Hattaf derivatives with non-singular kernels. The associated fractional integral of this new mixed fractional derivative is rigorously defined. A numerical scheme based on Lagrange polynomial interpolation is developed to approximate solutions of fractional differential equations (FDEs) involving the mixed fractional derivative. The scheme includes recent numerical methods and is shown to be effective and rapidly converging. An application in computational biology is presented, where the new mixed fractional derivative is used to model the evolution of a cell population in the human body. The study highlights the advantages of the new mixed fractional derivative, such as its non-locality and flexibility, and suggests future research directions, including the development of a general theory and the derivation of new fractal-fractional operators.This study introduces a new mixed fractional derivative that encompasses various types of fractional derivatives, including the Riemann–Liouville and Caputo derivatives with singular kernels, and the Caputo–Fabrizio, Atangana–Baleanu, and generalized Hattaf derivatives with non-singular kernels. The associated fractional integral of this new mixed fractional derivative is rigorously defined. A numerical scheme based on Lagrange polynomial interpolation is developed to approximate solutions of fractional differential equations (FDEs) involving the mixed fractional derivative. The scheme includes recent numerical methods and is shown to be effective and rapidly converging. An application in computational biology is presented, where the new mixed fractional derivative is used to model the evolution of a cell population in the human body. The study highlights the advantages of the new mixed fractional derivative, such as its non-locality and flexibility, and suggests future research directions, including the development of a general theory and the derivation of new fractal-fractional operators.