To derive Euler's method, recall how the derivative
is computed. By definition, the derivative of 
is
From the differential equation, this limit is functionally given as f(t,y), i.e.
One way of deriving Euler's method comes about by considering the quotient in this limit to be a good approximation when h is small,
Euler's method is derived from the above equation
by asserting equality in this approximation and replacing h with
the step size 
, y with the approximation 
.
Rearranging terms yields the recursion relationship
which is Euler's method. Equation (5.11)
builds the approximation at each step using the previous values of time
and
. For the initial approximation, 
is used or some close approximation. With this initialization, 
can be computed, then
may be used to compute 
, and so on.
The next section analyzes the error associated with
Euler's method. As a motivation for the aspects of error analysis, the
preceding example is presented with a smaller step size in t. Note
the differences in the error in the computations of Example 5.4
where 
and Example 5.5 where this step size is halved.
