@(@\newcommand{\W}[1]{ \; #1 \; }
\newcommand{\R}[1]{ {\rm #1} }
\newcommand{\B}[1]{ {\bf #1} }
\newcommand{\D}[2]{ \frac{\partial #1}{\partial #2} }
\newcommand{\DD}[3]{ \frac{\partial^2 #1}{\partial #2 \partial #3} }
\newcommand{\Dpow}[2]{ \frac{\partial^{#1}}{\partial {#2}^{#1}} }
\newcommand{\dpow}[2]{ \frac{ {\rm d}^{#1}}{{\rm d}\, {#2}^{#1}} }@)@This is cppad-20221105 documentation. Here is a link to its
current documentation
.
Exponential Function Forward Mode Theory
Derivatives
If @(@
F(x)
@)@ is @(@
\R{exp} (x)
@)@ or @(@
\R{expm1} (x)
@)@
the corresponding derivative satisfies the equation
@[@
1 * F^{(1)} (x) - 1 * F (x)
=
d^{(0)}
=
\left\{ \begin{array}{ll}
0 & \R{if} \; F(x) = \R{exp}(x)
\\
1 & \R{if} \; F(x) = \R{expm1}(x)
\end{array} \right.
@]@
where the equation above defines @(@
d^{(0)}
@)@.
In the
standard math function differential equation
,
@(@
A(x) = 1
@)@,
@(@
B(x) = 1
@)@,
and @(@
D(x) = d^{(0)}
@)@.
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.
