Proposition of a Recursive Formula to Calculate the Higher Order Derivative of a Composite Function without Using the Resolution of the Diophantine Equation

El Khomssi Mohammed; Chaachoui Ghizlane; Ez-Zriouli Rachid

El Khomssi Mohammed Modelling and Scientic Computing Laboratory, Faculty of Science and Technology of Fez, Box 2202, University S. M. Ben Abdellah Fez, Morocco.
Chaachoui Ghizlane Modelling and Scientic Computing Laboratory, Faculty of Science and Technology of Fez, Box 2202, University S. M. Ben Abdellah Fez, Morocco.
Ez-Zriouli Rachid Modelling and Scientic Computing Laboratory, Faculty of Science and Technology of Fez, Box 2202, University S. M. Ben Abdellah Fez, Morocco.

Keywords: Higher order derivative, composite function, diophantine equation, Faa Di Bruno, recurcive formula

Abstract

The formula of Fa`a Di Bruno provides a powerful tool to calculate the higher order derivative of a composite function. Unfortunately it has three weaknesses: it is not a recursive formula, it totally depends on the resolution of the diophantine equation and a change in the order of the derivative requires the total change of the calculation. With these weaknesses and the absence of a formula to program, Fa`a Di Brunos formula is less useful for formal computation.
Other complicated techniques based on ?nite di?erence calculation (see [ 1]) are recursive, however the complexity of the calculation algorithm is very high. There is as well some techniques based on graphs (see [2]) to calculate the coe?cients to a certain order, but without giving the general formula.
In our work we propose a new formula to calculate the higher order derivative of a composite function gof. It is of great interest, because it is recursive and it is not based on the resolution of the diophantine equation. We complete this work by giving an expression that allows to ?nd directly the n-th derivative of a composite function.