## ABSTRACT

This is the simplest form of “finite difference derivative” and is called the “forward” difference approximation to the derivative. E(x) represents the “error” in the approximation. In order to estimate the size of the error term, consider the Taylor expansion of f(x + Δx) in the neighborhood of x:

f x x f x xf x

x f x

x f x( ) ( ) ( )

! ( )

! ( )+ = + ′ + ′′ + ′′′ +∆ ∆

∆ ∆ ∆2 3

2 3 x

f xiv 4

4! ( ) + (4.2)

Equation 4.1 results from truncating all but the first two terms in the Taylor expansion of Equation 4.2. In order to determine how good the approximation is, consider temporarily retaining the term involving f .ʹ This gives

′ =

+ − − ′′ +f x

f x x f x x

x f x( )

( ) ( ) !