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@(@\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 .
The Pow Integer Exponent: Example and Test

# include <cppad/cppad.hpp>
# include <cmath>

bool pow_int(void)
{   bool ok = true;
    using CppAD::AD;
    using CppAD::NearEqual;
    double eps99 = 99.0 * std::numeric_limits<double>::epsilon();

    // declare independent variables and start tape recording
    size_t n  = 1;
    double x0 = -0.5;
    CPPAD_TESTVECTOR(AD<double>) x(n);
    x[0]      = x0;
    CppAD::Independent(x);

    // dependent variable vector
    size_t m = 7;
    CPPAD_TESTVECTOR(AD<double>) y(m);
    for(size_t i = 0; i < m; i++)
        y[i] = CppAD::pow(x[0], int(i) - 3);

    // create f: x -> y and stop tape recording
    CppAD::ADFun<double> f(x, y);

    // check value
    double check;
    for(size_t i = 0; i < m; i++)
    {   check = std::pow(x0, double(i) - 3.0);
        ok &= NearEqual(y[i] , check,  eps99 , eps99);
    }

    // forward computation of first partial w.r.t. x[0]
    CPPAD_TESTVECTOR(double) dx(n);
    CPPAD_TESTVECTOR(double) dy(m);
    dx[0] = 1.;
    dy    = f.Forward(1, dx);
    for(size_t i = 0; i < m; i++)
    {   check = (double(i) - 3.0) * std::pow(x0, double(i) - 4.0);
        ok &= NearEqual(dy[i] , check,  eps99 , eps99);
    }

    // reverse computation of derivative of y[i]
    CPPAD_TESTVECTOR(double)  w(m);
    CPPAD_TESTVECTOR(double) dw(n);
    for(size_t i = 0; i < m; i++)
        w[i] = 0.;
    for(size_t i = 0; i < m; i++)
    {   w[i] = 1.;
        dw    = f.Reverse(1, w);
        check = (double(i) - 3.0) * std::pow(x0, double(i) - 4.0);
        ok &= NearEqual(dw[0] , check,  eps99 , eps99);
        w[i] = 0.;
    }

    return ok;
}
Input File: example/utility/pow_int.cpp