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The derivatives of eight Horn hypergeometric functions [four Appell and four (degenerate) confluent] with respect to their parameters are studied. The first derivatives are expressed, systematically, as triple infinite summations or, alternatively, as single summations of two-variable Kampé de Fériet functions. Taking advantage of previously established expressions for the derivative of the confluent or Gaussian hypergeometric functions, the generalizationto the nth derivative of Horn?s functions with respect to their parameters is rather straightforward in most cases; the results are expressed in terms of n + 2 infinitesummations. Following a similar procedure, mixed derivatives are also treated. Anillustration of the usefulness of the derivatives of F1, with respect to the first and thirdparameters, is given with the study of autoionization of atoms occurring as part ofa post-collisional process. Their evaluation setting the Coulomb charge to zero providesthe coefficients of a Born-like expansion of the interaction.