% 2nd order accurate first derivative of a function 
% u(x) data found in the file: data_MoreReal.dat

  load data_MoreReal.dat
  x = data_MoreReal(:,1);
  u = data_MoreReal(:,2);
  dx=(x(end)-x(1))/(length(x)-1);
 
  n=length(x); % number of points

  % Use second order forward difference
  ux(1)=(-3*u(1)+4*u(2)-u(3))/(2*dx);

  % Use second order center difference
  for j=2:n-1
    ux(j)=(u(j+1)-u(j-1))/(2*dx);
  end

  % Use second order backward difference
  ux(n)=(3*u(n)-4*u(n-1)+u(n-2))/(2*dx);

  % plot the numerical integral
  figure(1), plot(x,ux,'r*');

  % plot the exact integral (this is known because I obtained
  % the data in data_MoreReal using: -tanh(x).*sech(x)
  figure(1), hold on, plot(x,-sech(x).*tanh(x),'b');