A mathematical function, specified by a computer program, can be differentiated efficiently through the exploitation of its program structure. The important properties of a program for the efficient generation of derivative code are the asymmetries between the number of inputs and outputs of program components at various levels of abstraction and the mathematical complexity of the involved operators. Automatic generation of efficient derivative codes thus requires analysis of programs for detection of such properties and systematic methods for their exploitation in composing the derivative codes. We suggest a hierarchical approach based on a partitioning of the computational or program graph as a means to deduce workable solutions to this hard problem. Each partition corresponds to a localized scope for derivative computation, and hierarchical partitions provide a mechanism for exploiting program structure at various levels. As a particular example, we discuss dynamic programming appro...