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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 AD Power Function: Example and Test

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

bool pow(void)
{   bool ok = true;

    using CppAD::AD;
    using CppAD::NearEqual;
    double eps = 10. * std::numeric_limits<double>::epsilon();

    // domain space vector
    size_t n  = 2;
    double x = 0.5;
    double y = 2.;
    CPPAD_TESTVECTOR(AD<double>) axy(n);
    axy[0]      = x;
    axy[1]      = y;

    // declare independent variables and start tape recording
    CppAD::Independent(axy);

    // range space vector
    size_t m = 3;
    CPPAD_TESTVECTOR(AD<double>) az(m);
    az[0] = CppAD::pow(axy[0], axy[1]); // pow(variable, variable)
    az[1] = CppAD::pow(axy[0], y);      // pow(variable, parameter)
    az[2] = CppAD::pow(x,     axy[1]);  // pow(parameter, variable)

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

    // check value
    double check = std::pow(x, y);
    size_t i;
    for(i = 0; i < m; i++)
        ok &= NearEqual(az[i] , check,  eps, eps);

    // forward computation of first partial w.r.t. x
    CPPAD_TESTVECTOR(double) dxy(n);
    CPPAD_TESTVECTOR(double) dz(m);
    dxy[0] = 1.;
    dxy[1] = 0.;
    dz    = f.Forward(1, dxy);
    check = y * std::pow(x, y-1.);
    ok   &= NearEqual(dz[0], check, eps, eps);
    ok   &= NearEqual(dz[1], check, eps, eps);
    ok   &= NearEqual(dz[2],    0., eps, eps);

    // forward computation of first partial w.r.t. y
    dxy[0] = 0.;
    dxy[1] = 1.;
    dz    = f.Forward(1, dxy);
    check = std::log(x) * std::pow(x, y);
    ok   &= NearEqual(dz[0], check, eps, eps);
    ok   &= NearEqual(dz[1],    0., eps, eps);
    ok   &= NearEqual(dz[2], check, eps, eps);

    // reverse computation of derivative of z[0] + z[1] + z[2]
    CPPAD_TESTVECTOR(double)  w(m);
    CPPAD_TESTVECTOR(double) dw(n);
    w[0]  = 1.;
    w[1]  = 1.;
    w[2]  = 1.;
    dw    = f.Reverse(1, w);
    check = y * std::pow(x, y-1.);
    ok   &= NearEqual(dw[0], 2. * check, eps, eps);
    check = std::log(x) * std::pow(x, y);
    ok   &= NearEqual(dw[1], 2. * check, eps, eps);

    // use a VecAD<Base>::reference object with pow
    CppAD::VecAD<double> v(2);
    AD<double> zero(0);
    AD<double> one(1);
    v[zero]           = axy[0];
    v[one]            = axy[1];
    AD<double> result = CppAD::pow(v[zero], v[one]);
    ok               &= NearEqual(result, az[0], eps, eps);

    return ok;
}
Input File: example/general/pow.cpp