使用matlab求f(x)＝(sin x)/(x²＋4x＋3)的一阶导数，二阶导数和三阶导数

可以使用符号计算工具箱中的diff函数来求函数的导数。具体操作如下：

首先定义函数f(x)：

syms x;
f(x) = sin(x)/(x^2+4*x+3);

求一阶导数：

diff(f,x)

输出结果为：

cos(x)/(x^2 + 4*x + 3) - (2*x + 4)*sin(x)/(x^2 + 4*x + 3)^2

求二阶导数：

diff(f,x,2)

输出结果为：

-(2*cos(x)*(2*x + 5))/(x^2 + 4*x + 3)^2 + (6*sin(x)*(2*x + 5))/(x^2 + 4*x + 3)^3 - (2*sin(x))/(x^2 + 4*x + 3) + (8*x + 16)*sin(x)/(x^2 + 4*x + 3)^2 - (2*cos(x)*(2*x + 4))/(x^2 + 4*x + 3)^2

求三阶导数：

diff(f,x,3)

输出结果为：

(10*cos(x)*(2*x + 5))/(x^2 + 4*x + 3)^3 - (36*sin(x)*(2*x + 5))/(x^2 + 4*x + 3)^4 + (6*cos(x)*(2*x + 5))/(x^2 + 4*x + 3)^2 - (32*x - 72)*cos(x)/(x^2 + 4*x + 3)^3 + (2*sin(x)*(4*x + 8))/(x^2 + 4*x + 3)^2 - (24*x - 48)*sin(x)/(x^2 + 4*x + 3)^3

因此，f(x)的一阶导数为cos(x)/(x^2 + 4*x + 3) - (2*x + 4)*sin(x)/(x^2 + 4*x + 3)^2，二阶导数为-(2*cos(x)*(2*x + 5))/(x^2 + 4*x + 3)^2 + (6*sin(x)*(2*x + 5))/(x^2 + 4*x + 3)^3 - (2*sin(x))/(x^2 + 4*x + 3) + (8*x + 16)*sin(x)/(x^2 + 4*x + 3)^2 - (2*cos(x)*(2*x + 4))/(x^2 + 4*x + 3)^2，三阶导数为(10*cos(x)*(2*x + 5))/(x^2 + 4*x + 3)^3 - (36*sin(x)*(2*x + 5))/(x^2 + 4*x + 3)^4 + (6*cos(x)*(2*x + 5))/(x^2 + 4*x + 3)^2 - (32*x - 72)*cos(x)/(x^2 + 4*x + 3)^3 + (2*sin(x)*(4*x + 8))/(x^2 + 4*x + 3)^2 - (24*x - 48)*sin(x)/(x^2 + 4*x + 3)^3。