It is shown hew matrix differential calculus can be used to obtain simplified expressions for higher-order terms that appear in standard approximate non-linear filtering algorithms such as the first-order bias-corrected filter, the truncated second-order filter, and the modified Gaussian second-order filter. These terms are much simpler and more compact than the standard expressions given in the literature; particularly, no element-by-element computations are necessary. For comparison, the approximate version of the Daum filter is also included.