 
Summary: NUMERICAL DIFFERENTIATION
There are two major reasons for considering numeri
cally approximations of the differentiation process.
1. Approximation of derivatives in ordinary differen
tial equations and partial differential equations.
This is done in order to reduce the differential
equation to a form that can be solved more easily
than the original differential equation.
2. Forming the derivative of a function f(x) which is
known only as empirical data {(xi, yi)  i = 1, . . . , m}.
The data generally is known only approximately,
so that yi f(xi), i = 1, . . . , m.
Recall the definition
f0(x) = lim
h0
f(x + h)  f(x)
h
This justifies using
f0(x)
f(x + h)  f(x)
