Computation of derivatives of output maps along solutions of nonlinear systems

The authors present two formulas pertaining to the output of a nonlinear system. In the first, they give an infinite product expansion for the output, enabling the Taylor series to be evaluated directly. In the second, they give an explicit formula for the nth-order total derivative. The common tool for these formulas is the shuffle product in an algebra of noncommuting variables. This product seems to be an integral part of any computational approach to evaluating coefficients in the expansions obtained