Let $X$ be a compact complex manifold, let $T_X$ denote its holomorphic tangent sheaf, and let $D$ be the analytic dual-number germ with local algebra $\mathbb C[\varepsilon]/(\varepsilon^2)$. The set of isomorphism classes of first-order analytic deformations of $X$ over $D$ is naturally in bijection with $H^1(X,T_X)$. Under this bijection, the product deformation corresponds to $0$, and the class attached to a deformation is represented by the Cech cocycle of holomorphic vector fields obtained from first-order transition functions in local product charts.