% 2nd order accurate first derivative of a function u(x) = sech(x)
% over a domain x = [-2,2]
dx=.4;

  x = (-2:dx:2);
  u = sech(x);
  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);

  % Compare results:
  ux_exact = -tanh(x).*sech(x);
  error_1norm = norm(ux_exact-ux,1)

  figure(1), plot(x,ux_exact,'b',x,ux,'r*');

  clear ux_exact ux