function value = runge_deriv2 ( x )

%*****************************************************************************80
%
%% runge_deriv2() evaluates the second derivative of the Runge function.
%
%  Licensing:
%
%    This code is distributed under the MIT license.
%
%  Modified:
%
%    23 March 2025
%
%  Author:
%
%    John Burkardt
%
%  Input:
%
%    real x: the argument of the function.
%
%  Output:
%
%    real value, the value of the second derivative.
%
  u = - 50.0 * x;
  up = - 50.0;
  v = ( 1.0 + 25.0 * x .* x ).^2;
  vp = 2.0 * ( 1.0 + 25.0 * x .* x ) .* ( 50.0 * x );

  value = ( up .* v - u .* vp ) ./ v.^2;

  return
end
