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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 .
Second Partials Reverse Driver: Example and Test
# include <cppad/cppad.hpp>
namespace { // -----------------------------------------------------
// define the template function in empty namespace
// bool RevTwoCases<BaseVector, SizeVector_t>(void)
template <class BaseVector, class SizeVector_t>
bool RevTwoCases()
{   bool ok = true;
    using CppAD::AD;
    using CppAD::NearEqual;
    double eps99 = 99.0 * std::numeric_limits<double>::epsilon();
    using CppAD::exp;
    using CppAD::sin;
    using CppAD::cos;

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

    // declare independent variables and starting recording
    CppAD::Independent(X);

    // a calculation between the domain and range values
    AD<double> Square = X[0] * X[0];

    // range space vector
    size_t m = 3;
    CPPAD_TESTVECTOR(AD<double>)  Y(m);
    Y[0] = Square * exp( X[1] );
    Y[1] = Square * sin( X[1] );
    Y[2] = Square * cos( X[1] );

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

    // new value for the independent variable vector
    BaseVector x(n);
    x[0] = 2.;
    x[1] = 1.;

    // set i and j to compute specific second partials of y
    size_t p = 2;
    SizeVector_t i(p);
    SizeVector_t j(p);
    i[0] = 0; j[0] = 0; // for partials y[0] w.r.t x[0] and x[k]
    i[1] = 1; j[1] = 1; // for partials y[1] w.r.t x[1] and x[k]

    // compute the second partials
    BaseVector ddw(n * p);
    ddw = f.RevTwo(x, i, j);

    // partials of y[0] w.r.t x[0] is 2 * x[0] * exp(x[1])
    // check partials of y[0] w.r.t x[0] and x[k] for k = 0, 1
    ok &=  NearEqual(      2.*exp(x[1]), ddw[0*p+0], eps99, eps99);
    ok &=  NearEqual( 2.*x[0]*exp(x[1]), ddw[1*p+0], eps99, eps99);

    // partials of y[1] w.r.t x[1] is x[0] * x[0] * cos(x[1])
    // check partials of F_1 w.r.t x[1] and x[k] for k = 0, 1
    ok &=  NearEqual(    2.*x[0]*cos(x[1]), ddw[0*p+1], eps99, eps99);
    ok &=  NearEqual( -x[0]*x[0]*sin(x[1]), ddw[1*p+1], eps99, eps99);

    return ok;
}
} // End empty namespace
# include <vector>
# include <valarray>
bool RevTwo(void)
{   bool ok = true;
        // Run with BaseVector equal to three different cases
        // all of which are Simple Vectors with elements of type double.
    ok &= RevTwoCases< CppAD::vector <double>, std::vector<size_t> >();
    ok &= RevTwoCases< std::vector   <double>, std::vector<size_t> >();
    ok &= RevTwoCases< std::valarray <double>, std::vector<size_t> >();

        // Run with SizeVector_t equal to two other cases
        // which are Simple Vectors with elements of type size_t.
    ok &= RevTwoCases< std::vector <double>, CppAD::vector<size_t> >();
    ok &= RevTwoCases< std::vector <double>, std::valarray<size_t> >();

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