Starting with Taylor's approximation,
or, coupling this with equation (5.1),
where
taken to be in the interval bounded by t and 
. Since f(t,y(t)) is given in the statement
of the problem, the derivatives 
are readily computable. The ``degree n'' Taylor series approximation
to the differential equation is obtained by omitting the remainder term, 
. By truncating the remainder, the resulting approximation is easily expressed,
but adds the additional problem-specific burden of calculating the
derivatives of f with respect to t. Since f is described
as a function of both t and y, derivatives of f with
respect to t will require use of the chain rule, which implies that
the total derivative of f with respect to t is computed as
General formulas for the higher-order derivatives of f are computed similarly, and increase significantly in their degree of complication. The complexity of computing the higher-order derivatives may be offset by the particular form that f(t,y(t)) takes in equation (5.1).
From the above derivation of Taylor methods, it is clear that one can utilize this approach in order to come up with a method which is accurate up to an arbitrarily-prescribed degree. However, the complexity found in computing the derivatives of the function f is often cumbersome. Additionally, it is common that the function f is not known explicitly, but is computable for any given point in the (t,y) plane. In this situation, calculation of the derivatives of f would need to be computed numerically, compounding the cumbersome nature of the implementation.
