Let C be a curve with Jacobian variety J defined over an arbitrary field k. In this paper, we show that the logarithmic derivative induces a natural homomorphism from the group J( k) of k-rational points on J into the group (H 1(C, O c) ⊗ k Ω k Z 1) δ(Γ(C, Ω C k 1)) , where δ is a connecting homomorphism in a natural sequence of Zariski cohomology groups. When C = E is an elliptic curve with j-invariant equal to j, we show that the image of δ is the k-vector subspace of Ω k z 1 spanned by the absolute differential dj. Thus, we can interpret the logarithmic derivative as a map dlog : E(k) → Ω k[j] z 1 . Finally, we compute the kernel of this morphism explicitly. To describe the main theorem, write the Weierstrass equation of E in the form y 2 = x 3 + a 4 x + a 6. Let k 0 be the prime field of k and let F be the algebraic closure in k of the field k 0( a 4, a 6). We show that the kernel of dlog can be identified with the group E( F) of F-rational points on E. In particular, notice that when k = C is the field of complex numbers, then the kernel of dlog is countable, and its image must be uncountable.