Derivatives
If @(@
F(x)
@)@ is @(@
\R{log} (x)
@)@ or @(@
\R{log1p} (x)
@)@
the corresponding derivative satisfies the equation
@[@
( \bar{b} + x ) * F^{(1)} (x) - 0 * F (x) = 1
@]@
where
@[@
\bar{b}
=
\left\{ \begin{array}{ll}
0 & \R{if} \; F(x) = \R{log}(x)
\\
1 & \R{if} \; F(x) = \R{log1p}(x)
\end{array} \right.
@]@
In the
standard math function differential equation
,
@(@
A(x) = 0
@)@,
@(@
B(x) = \bar{b} + x
@)@,
and @(@
D(x) = 1
@)@.
We use @(@
a
@)@, @(@
b
@)@, @(@
d
@)@,
and @(@
z
@)@ to denote the
Taylor coefficients for
@(@
A [ X (t) ]
@)@,
@(@
B [ X (t) ]
@)@,
@(@
D [ X (t) ]
@)@,
and @(@
F [ X(t) ]
@)@ respectively.