Many differentiation methods rely on the notion of "primitive" differentiation rules that can be composed via various formulations of the chain rule. Using DiffRules, you can define new differentiation rules, query whether or not a given rule exists, and symbolically apply rules to simple Julia expressions.
Note that DiffRules is not a fully-fledged symbolic differentiation tool. It is a (very) simple global database of common derivative definitions, and was developed with the goal of improving derivative coverage in downstream tools.