使用matlab求f(x)=(sin x)/(x²+4x+3),f(x)的一阶导数,二阶导数和三阶导数
时间: 2024-03-21 19:40:48 浏览: 116
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。
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