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Description: Maximum likelihood estimation of generalized linear mixed models (GLMMs) is difficult due to marginalization of the random effects. Com- puting derivatives of a fitted GLMM’s likelihood is also difficult, especially because the derivatives are not by-products of popular estimation algo- rithms. In this paper, we describe GLMM derivatives along with a quadra- ture method to efficiently compute them, focusing on lme4 models with a single clustering variable. We describe how psychometric results related to IRT are helpful for obtaining these derivatives, as well as for verifying the derivatives’ accuracies. After describing the derivative computation meth- ods, we illustrate the many possible uses of these derivatives, including ro- bust standard errors, score tests of fixed effect parameters, and likelihood ratio tests of non-nested models. The derivative computation methods and applications described in the paper are all available in easily-obtained R packages.