Approximation of a Derivative in One Variable

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

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/. This gives 

The last term is an estimate of Eix), which is proportional to Ax. A terminology used to describe this is the Big O notation or the order of magnitude notation. The approximation of Equation 4.1 is, therefore, 0(Ax), or the error in the approximation is proportional to Ax. 

This is called the backward difference approximation to the first derivative and is also C (Ax). 

Note that the error is proportional to Ax, or it is O(Ax ). Equations 4.1 and 4.4 are first-order correct, and Equation 4.6 is second-order correct. Equation 4.6 (without the error term) is called the central difference approximation to the first derivative. Adding Equations 4.2 and 4.3 gives the following result  

---