matlab 求解 f(x)=x(x+1)(x+2)(x+3)(x+4)...(x+n-2)(x+n-1)(x+n)的导数

问题的提出

代码求解

	clc; clear; close all;
    filename='how10.txt';
    syms x; % be careful performance issue.
    fileID = fopen(filename, 'w');
    str="x";
    f=str2sym(str);
    fprintf(fileID,'%s\r\n',string(f));
    result=diff(f,x);
    fprintf(fileID,'%s\r\n',string(result));
    ff=matlabFunction(result);
    fprintf(fileID,'%d\r\n',1);
    fprintf(fileID,'\r\n');
    for i = 1:10
        newstr="*(x+"+int2str(i)+")";
        str=str+newstr;
        f=str2sym(str);
        fprintf(fileID,'%s\r\n',string(f));
        result=diff(f,x);
        fprintf(fileID,'%s\r\n',string(result));
        ff=matlabFunction(result);
        fprintf(fileID,'%f\r\n',ff(0));
        fprintf(fileID,'\r\n');
    end
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代码输出

x 原函数 $ f(x) $
1 导函数 !$ f(x)^{'} $
1 !$ f(0)^{'} $

x*(x + 1) 原函数 !$ f(x) $
2*x + 1 导函数 !$ f(x)^{'} $
1.000000 !$ f(0)^{'} $

x*(x + 1)(x + 2)
(x + 1)(x + 2) + x*(x + 1) + x*(x + 2)
2.000000

x*(x + 1)(x + 2)(x + 3)
x*(x + 1)(x + 2) + x(x + 1)(x + 3) + x(x + 2)(x + 3) + (x + 1)(x + 2)*(x + 3)
6.000000

x*(x + 1)(x + 2)(x + 3)(x + 4)
x(x + 1)(x + 2)(x + 3) + x*(x + 1)(x + 2)(x + 4) + x*(x + 1)(x + 3)(x + 4) + x*(x + 2)(x + 3)(x + 4) + (x + 1)(x + 2)(x + 3)*(x + 4)
24.000000

x*(x + 1)(x + 2)(x + 3)(x + 4)(x + 5)
x*(x + 1)(x + 2)(x + 3)(x + 4) + x(x + 1)(x + 2)(x + 3)(x + 5) + x(x + 1)(x + 2)(x + 4)(x + 5) + x(x + 1)(x + 3)(x + 4)(x + 5) + x(x + 2)(x + 3)(x + 4)(x + 5) + (x + 1)(x + 2)(x + 3)(x + 4)*(x + 5)
120.000000

x*(x + 1)(x + 2)(x + 3)(x + 4)(x + 5)(x + 6)
x(x + 1)(x + 2)(x + 3)(x + 4)(x + 5) + x*(x + 1)(x + 2)(x + 3)(x + 4)(x + 6) + x*(x + 1)(x + 2)(x + 3)(x + 5)(x + 6) + x*(x + 1)(x + 2)(x + 4)(x + 5)(x + 6) + x*(x + 1)(x + 3)(x + 4)(x + 5)(x + 6) + x*(x + 2)(x + 3)(x + 4)(x + 5)(x + 6) + (x + 1)(x + 2)(x + 3)(x + 4)(x + 5)*(x + 6)
720.000000

x*(x + 1)(x + 2)(x + 3)(x + 4)(x + 5)(x + 6)(x + 7)
(x + 1)(x + 2)(x + 3)(x + 4)(x + 5)(x + 6)(x + 7) + x*(x + 1)(x + 2)(x + 3)(x + 4)(x + 5)(x + 6) + x(x + 1)(x + 2)(x + 3)(x + 4)(x + 5)(x + 7) + x(x + 1)(x + 2)(x + 3)(x + 4)(x + 6)(x + 7) + x(x + 1)(x + 2)(x + 3)(x + 5)(x + 6)(x + 7) + x(x + 1)(x + 2)(x + 4)(x + 5)(x + 6)(x + 7) + x(x + 1)(x + 3)(x + 4)(x + 5)(x + 6)(x + 7) + x(x + 2)(x + 3)(x + 4)(x + 5)(x + 6)*(x + 7)
5040.000000

x*(x + 1)(x + 2)(x + 3)(x + 4)(x + 5)(x + 6)(x + 7)(x + 8)
(x + 1)(x + 2)(x + 3)(x + 4)(x + 5)(x + 6)(x + 7)(x + 8) + x*(x + 1)(x + 2)(x + 3)(x + 4)(x + 5)(x + 6)(x + 7) + x*(x + 1)(x + 2)(x + 3)(x + 4)(x + 5)(x + 6)(x + 8) + x*(x + 1)(x + 2)(x + 3)(x + 4)(x + 5)(x + 7)(x + 8) + x*(x + 1)(x + 2)(x + 3)(x + 4)(x + 6)(x + 7)(x + 8) + x*(x + 1)(x + 2)(x + 3)(x + 5)(x + 6)(x + 7)(x + 8) + x*(x + 1)(x + 2)(x + 4)(x + 5)(x + 6)(x + 7)(x + 8) + x*(x + 1)(x + 3)(x + 4)(x + 5)(x + 6)(x + 7)(x + 8) + x*(x + 2)(x + 3)(x + 4)(x + 5)(x + 6)(x + 7)(x + 8)
40320.000000

x*(x + 1)(x + 2)(x + 3)(x + 4)(x + 5)(x + 6)(x + 7)(x + 8)(x + 9)
x*(x + 1)(x + 2)(x + 3)(x + 4)(x + 5)(x + 6)(x + 7)(x + 8) + x(x + 1)(x + 2)(x + 3)(x + 4)(x + 5)(x + 6)(x + 7)(x + 9) + x(x + 1)(x + 2)(x + 3)(x + 4)(x + 5)(x + 6)(x + 8)(x + 9) + x(x + 1)(x + 2)(x + 3)(x + 4)(x + 5)(x + 7)(x + 8)(x + 9) + x(x + 1)(x + 2)(x + 3)(x + 4)(x + 6)(x + 7)(x + 8)(x + 9) + x(x + 1)(x + 2)(x + 3)(x + 5)(x + 6)(x + 7)(x + 8)(x + 9) + x(x + 1)(x + 2)(x + 4)(x + 5)(x + 6)(x + 7)(x + 8)(x + 9) + x(x + 1)(x + 3)(x + 4)(x + 5)(x + 6)(x + 7)(x + 8)(x + 9) + x(x + 2)(x + 3)(x + 4)(x + 5)(x + 6)(x + 7)(x + 8)(x + 9) + (x + 1)(x + 2)(x + 3)(x + 4)(x + 5)(x + 6)(x + 7)(x + 8)*(x + 9)
362880.000000

x*(x + 1)(x + 2)(x + 3)(x + 4)(x + 5)(x + 6)(x + 7)(x + 8)(x + 9)(x + 10)
x(x + 1)(x + 2)(x + 3)(x + 4)(x + 5)(x + 6)(x + 7)(x + 8)(x + 9) + x*(x + 1)(x + 2)(x + 3)(x + 4)(x + 5)(x + 6)(x + 7)(x + 8)(x + 10) + x*(x + 1)(x + 2)(x + 3)(x + 4)(x + 5)(x + 6)(x + 7)(x + 9)(x + 10) + x*(x + 1)(x + 2)(x + 3)(x + 4)(x + 5)(x + 6)(x + 8)(x + 9)(x + 10) + x*(x + 1)(x + 2)(x + 3)(x + 4)(x + 5)(x + 7)(x + 8)(x + 9)(x + 10) + x*(x + 1)(x + 2)(x + 3)(x + 4)(x + 6)(x + 7)(x + 8)(x + 9)(x + 10) + x*(x + 1)(x + 2)(x + 3)(x + 5)(x + 6)(x + 7)(x + 8)(x + 9)(x + 10) + x*(x + 1)(x + 2)(x + 4)(x + 5)(x + 6)(x + 7)(x + 8)(x + 9)(x + 10) + x*(x + 1)(x + 3)(x + 4)(x + 5)(x + 6)(x + 7)(x + 8)(x + 9)(x + 10) + x*(x + 2)(x + 3)(x + 4)(x + 5)(x + 6)(x + 7)(x + 8)(x + 9)(x + 10) + (x + 1)(x + 2)(x + 3)(x + 4)(x + 5)(x + 6)(x + 7)(x + 8)(x + 9)*(x + 10)
3628800.000000

分析代码结果

可以看出来
结果为!$ n! $