Nth Derivatives(In Greek) (original) (raw)

A SERIES FORMULA FOR OBTAINING THE ORDER DERIVATIVE OF THE FUNCTIONS OF THE FORM

We generated a series method for computing the n th order derivative of a function that depends on five Independent variables by using the Leibnitz's theorem in the product rule. A Theorem that establishes the new method and its proof are presented also by using Mathematical Induction. The new series method does not require the knowledge of the preceding derivative before obtaining the Succeeding ones. 1.0 INTRODUCTION We desire the nth order derivative of the function; * () () () () ()+ () It follows that it's first derivative is given as; () Such that () () () and () () {(() ()+ * () () ()} () By using Leibnitz's theorem on the derivatives in (3) *(() ()+ {∑ (()) (())} () For * () () ()+ () , () ()-() () , ()-()

Explicit Formula for the n-th Derivative of a Quotient

2021

Leibniz’s rule for the n-th derivative of a product is a very well known and extremely useful formula. In this article, we introduce an analogous explicit formula for the n-th derivative of a quotient of two functions. Later, we use this formula to derive new partition identities and to develop expressions for some special n-th derivatives.

Differentiation and Integration 4.1 INTRODUCTION

Given a function f (x) explicitly or defined at a set of n + 1 distinct tabular points, we discuss methods to obtain the approximate value of the rth order derivative f (r) (x), r ≥ 1, at a tabular or a non-tabular point and to evaluate w x a b () z f (x) dx, where w(x) > 0 is the weight function and a and / or b may be finite or infinite. 4.2 NUMERICAL DIFFERENTIATION Numerical differentiation methods can be obtained by using any one of the following three techniques : (i) methods based on interpolation, (ii) methods based on finite differences, (iii) methods based on undetermined coefficients. Methods Based on Interpolation Given the value of f (x) at a set of n + 1 distinct tabular points x 0 , x 1 , ..., x n , we first write the interpolating polynomial P n (x) and then differentiate P n (x), r times, 1 ≤ r ≤ n, to obtain P n r () (x). The value of P n r () (x) at the point x*, which may be a tabular point or a non-tabular point gives the approximate value of f (r) (x) at the point x = x*. If we use the Lagrange interpolating polynomial P n (x) = l x f x i i i n () () = ∑ 0 (4.1) having the error term E n (x) = f (x) – P n (x) = () () ... () ()! x x x x x x n n − − − + 0 1 1 f (n+1) (ξ) (4.2) we obtain f (r) (x *) ≈ P x n r () () * , 1 ≤ r ≤ n and E x n r () () * = f (r) (x *) – P x n r () () * (4.3) is the error of differentiation. The error term (4.3) can be obtained by using the formula 212