With the definition of discrete lines introduced by Réveillès [REV91], there has been a wide range of research in discrete geometry and more precisely on the study of discrete lines. By the use of the linear time segment recognition algorithm of Debled and Réveillès [DR94], Vialard [VIA96a] has proposed a O(l) algorithm for computing the tangent in one point of a discrete curve where l is the average length of the tangent. By applying her algorithm to n points of a discrete curve, the complexity becomes O(n.l). This paper proposes a new approach for computing the tangent. It is based on a precise study of the tangent evolution along a discrete curve. The resulting algorithm has a O(n) complexity and is thus optimal. Some applications in curvature computation and a tombstones contours study are also presented.