In this paper, we propose an improved interpolation error estimate based on a new Taylor-like formula, which we apply to the finite element method. We first present a new first-order and second-order expansion formula with a reduced remainder. Then, we derive a new interpolation error estimate in \(W^{1,p}\) . We compare this with the classical error estimates based on the standard Taylor formula and the corresponding interpolation error estimate derived from the mean value theorem. We illustrate, with examples, the significant reduction this yields in finite element computation costs.