Effect of dx on error

Plot shows that error in fourth order is quite less as compared to the second and first order.

My Legends were not coming due to some software bugs,I will try to fix it by downloading it

clc
close all
clear all

dx=pi/40:pi/1000:pi/20

for i=1:length(dx)
    error1(i)=errorfds(dx(i));
    error2(i)=errorcds(dx(i));
    error3(i)=errorfourth(dx(i));
end

hold on
plot(dx,error1,\'r\') % red line is the first order derivative
plot(dx,error2,\'y\')  % yellow line is the second order derivative
plot(dx,error3,\'b\')  % blue line is the fourth order derivative

function [error_fds]= errorfds(dx)

%function=sin(x)/x^3; 
x=pi/3;

%analytical derivative=(x^3*cos(x)-sin(x)*3*x^2)/x^6;

analytical_derivative=(x^3*cos(x)-sin(x)*3*x^2)/x^6;

%numerical derivative

 %spacing between the grid points

%first order approximation
%forward differencing scheme (FDS)

forward_differencing=(sin(x+dx)/(x+dx)^3-sin(x)/x^3)/dx;
error_fds=abs(forward_differencing-analytical_derivative);

end

function [error_cds]= errorcds(dx)

%function=sin(x)/x^3; 
x=pi/3;

%analytical derivative=(x^3*cos(x)-sin(x)*3*x^2)/x^6;

analytical_derivative=(x^3*cos(x)-sin(x)*3*x^2)/x^6;

%numerical derivative

 %spacing between the grid points




%second order approximation-Central differencing scheme(CDS)

central_differencing=(sin(x+dx)/(x+dx)^3-sin(x-dx)/(x-dx)^3)/(2*dx);
error_cds=abs(central_differencing-analytical_derivative);


end

function [error_fourth_order_cds]= errorfourth(dx)

%function=sin(x)/x^3; 
x=pi/3;

%analytical derivative=(x^3*cos(x)-sin(x)*3*x^2)/x^6;

analytical_derivative=(x^3*cos(x)-sin(x)*3*x^2)/x^6;

%numerical derivative

 %spacing between the grid points






%3.fourth order approximation-Central differencing scheme(CDS)

F_i_minus_2=(sin(x-dx-dx)/(x-dx-dx)^3);
F_i_minus_1=8*(sin(x-dx)/(x-dx)^3);
F_i_plus_1=8*(sin(x+dx)/(x+dx)^3);
F_i_plus_2=(sin(x+dx+dx)/(x+dx+dx)^3);

fourth_order_approximation=(F_i_minus_2-F_i_minus_1+F_i_plus_1-F_i_plus_2)/(12*dx);
error_fourth_order_cds=abs(fourth_order_approximation-analytical_derivative);
end