Summary: Explicit functional forms for the generator derivatives of well-known one-parameter Archimedean copulas are derived. These derivatives are essential for likelihood inference as they appear in the copula density, conditional distribution functions, and the Kendall distribution function. They are also required for several asymmetric extensions of Archimedean copulas such as Khoudraji-transformed Archimedean copulas. Availability of the generator derivatives in a form that permits fast and accurate computation makes maximum-likelihood estimation for Archimedean copulas feasible, even in large dimensions. It is shown, by large scale simulations of the performance of maximum likelihood estimators under known margins, that the root mean squared error actually decreases with both dimension and sample size at a similar rate. Confidence intervals for the parameter vector are derived under known margins. Moreover, extensions to multi-parameter Archimedean families are given. All presented methods are implemented in the R package nacopula and can thus be studied in detail.