On the complexity of the computation of differentials and gradients

We obtain bounds for the complexity of circuit realisation of the system of differentials of orders from one to $k$ of an arbitrary elementary function in terms of the circuit complexity of this function. Similar bounds are obtained for the complexities of realisation of the Jacobian and Hessian matrices. We point out some applications to deduction of bounds for complexities of polynomials in several variables, linear transformations, and quadratic forms.
This research was supported by the Russian Foundation for Basic Research, grants 02–01–10142 and 02–01–00985, and by the Program of the President of the Russian Federation for supporting the leading scientific schools, grant 1807.2003.1.