Finite Difference Approximations of the First Derivative of a Function

The first derivative of a function is defined as

.

The simplest finite difference formulas for the first derivative of a function are:

(forward difference)
(central difference)
(backward difference)

Both forward and backward difference formulas have error , while the central difference formula has error .

In this Demonstration, we show the difference in values calculated from the three difference formulas and the exact value.

Questions:
1. Does the true error increase proportionally with the step size, , for the forward and backward difference formulas?
2. Does the true error increase proportionally with the square of the step size, , for the central difference formula?

Reference

[1] A. A. Kaw, D. Nguyen, and E. E. Kalu, Numerical Methods with Applications. http://numericalmethods.eng.usf.edu/publications_book.html.