In one conventional notation, a vertical bar placed after the index indicates differentiation, while a double bar indicates covariant differentiation (in another notation, the comma and semi-colon are used, but as these are special symbols to lisp and REDUCE they cannot be used here). In REDTEN the differentiation symbols are naturally | and ||. The operation is indicated by placing these symbols after the normal object indices (i.e. after the number of normal index elements equal to the rank of the object) and adjacent to the following index element, for example: g[a,b,|c]. The comma preceeding the | must not be left out, or the parser will become confused (a comma after the | is harmless).
Any index with more elements than the rank of the object must indicate a differentiation with the first extra element. The remaining elements are taken to also be derivative indices, and do not require a differentiation symbol (unless the type of differentiation changes). Covariant differentiation is always done one index-element at a time (i.e. intermediate objects are created); it also requires the existence of the Christoffel symbols (which will be created from the tensor metric if need be), see §4.2. An ordinary derivative of an indexed object is computed by taking the derivative of each component with respect to each of the coordinate names stored on the coords!* variable, see §2.4.1. In either case a new object is created to store the derivatives, see §4.1 and §4.2.