numerically and analytically solved solution

clear all;
close all;
clc;

x=linspace(pi/3,pi/2,10);
for i=1:10
analytical_derivative(i)=((x(i)^3.*cos(x(i)))-(sin(x(i)).*3.*x(i).^2))/x(i).^6;
end

dx=pi/40;

%first order approximation
forward_differincing =((sin(x+dx)/(x+dx).^3)-(sin(x)/(x).^3))/dx;

%second order approximation
central_differincing =((sin(x+dx)/(x+dx).^3)-(sin(x-dx)/(x-dx).^3))/(2*dx);

% fourth_order approximation
fourth_order_approximation=((sin(x-(2*dx))/(x-(2*dx)).^3)-(8*sin(x-dx)/(x-dx).^3)+(8*sin(x+dx)/(x+dx).^3)-(sin(x-(2*dx))/(x-(2*dx)).^3))/(12*dx);

error_first_order = abs(forward_differincing-analytical_derivative);

second_order = abs(central_differincing-analytical_derivative);

error_fourth_order = abs(fourth_order_approximation-analytical_derivative);

axis([-2 2 -8 -2]);
plot(error_first_order,analytical_derivative );
hold on;
plot (second_order,analytical_derivative);
hold on;
plot (error_fourth_order,analytical_derivative);
hold on;
xlabel(\'numerical\');
ylabel(\'analytical solution\');
legend(\'numerical\',\'analytical solution\');