Let K be a field of characteristic 0 and let W1 be the Lie algebra of the derivations of the polynomial ring K[t]. The algebra W1 admits a natural Z-grading. We describe the graded identities of W1 for this grading. It turns out that all these Z-graded identities are consequences of a collection of polynomials of degree 1, 2 and 3 and that they do not admit a finite basis. Recall that the "ordinary" (non-graded) identities of W1 coincide with the identities of the Lie algebra of the vector fields on the line and it is a long-standing open problem to find a basis for these identities. We hope that our paper might be a step to solving this problem.427 226251Bahturin, Y., (1987) Identical Relations in Lie Algebras, , first English ed., VNU Science Press BV, UtrechtBahturin, Y., Kochetov, M., Classification of group gradings on simple Lie algebras of types A, B, C and D (2010) J. Algebra, 324, pp. 2971-2989Bahturin, Y., Zaicev, M., Graded algebras and graded identities (2003) Lect. Notes Pure Appl. Math., 235, pp. 101-139. , M. Dekker, Polynomial Identities and Combinatorial MethodsElduque, A., Kochetov, M., Gradings on Simple Lie Algebras (2013) Math. Surveys Monogr., 189. , Amer. Math. Soc., Providence, RI, Atlantic Association for Research in the Math. Sciences (AARMS), Halifax, NSGiambruno, A., Souza, M.S., Graded polynomial identities and Specht property of the Lie algebra sl2 (2013) J. Algebra, 389, pp. 6-22Giambruno, A., Souza, M.S., Minimal varieties of graded Lie algebras of exponential growth and the special Lie algebra sl2 (2014) J. Pure Appl. Algebra, 218, pp. 1517-1527Giambruno, A., Zaicev, M., Polynomial Identities and Asymptotic Methods (2005) Math. Surveys Monogr., 122. , Amer. Math. Soc., Providence, RIKanel-Belov, A., Rowen, L., Computational Aspects of Polynomial Identities (2005) Res. Notes Math., 9. , A. K. Peters, Ltd., Wellesley, MAKemer, A., Ideals of Identities of Associative Algebras (1991) Transl. Math. Monogr., 87. , Amer. Math. Soc., Providence, RIKoshlukov, P., Graded polynomial identities for the Lie algebra sl2(K) (2008) Internat. J. Algebra Comput., 18, pp. 825-836Koshlukov, P., Krasilnikov, A., Silva, D.D.P., Graded identities for Lie algebras (2009) Contemp. Math., 499, pp. 181-188. , Amer. Math. SocKoshlukov, P., Krasilnikov, A., Just nonfinitely based varieties of 2-graded Lie algebras submitted for publicationMishchenko, S.P., Solvable subvarieties of a variety generated by a Witt algebra (1989) Math. USSR Sb., 64, pp. 415-426Razmyslov, Y., Identities of Algebras and Their Representations (1994) Transl. Math. Monogr., 138. , Amer. Math. Soc., Providence, RIRepin, D.V., Graded identities of a simple three-dimensional Lie algebra (2004) Vestn. Samar. Gos. Univ. Estestvennonauchn. Ser., (2), pp. 5-16. , Special Issue (in Russian)