完整推导导数公式体系（2）

2022-02-06 08:24 作者:匆匆-cc 0人读过 | 我要投稿

## 接上文

模块五：正弦函数

$f(x)%3D%5Csin%20x$

$%5Cbegin%7Balign%7D%0Af'(x)%26%3D%5Clim_%7B%5CDelta%20x%20%5Cto%200%7D%20%5Cfrac%7Bf(x%2B%5CDelta%20x)-f(x)%7D%7B%5CDelta%20x%7D%20%0A%5C%5C%26%3D%5Clim_%7B%5CDelta%20x%20%5Cto%200%7D%20%5Cfrac%7B%5Csin(x%2B%5CDelta%20x)-%5Csin%20x%7D%7B%5CDelta%20x%7D%20%0A%5C%5C%26%3D%5Clim_%7B%5CDelta%20x%20%5Cto%200%7D%5Cfrac%7B2%5Ccos%20%5Cleft(x%2B%5Cfrac%7B%5CDelta%20x%7D%7B2%7D%5Cright)%5Csin%5Cfrac%7B%5CDelta%20x%7D%7B2%7D%7D%7B%5CDelta%20x%7D%0A%5C%5C%26%3D%5Clim_%7B%5CDelta%20x%20%5Cto%200%7D%5Cfrac%7B2%5Ccos%20%5Cleft(x%2B%5Cfrac%7B%5CDelta%20x%7D%7B2%7D%5Cright)%5Cfrac%7B%5CDelta%20x%7D%7B2%7D%7D%7B%5CDelta%20x%7D%5C%23%0A%5C%5C%26%3D%5Clim_%7B%5CDelta%20x%5Cto0%7D%5Ccos%5Cleft(x%2B%5Cfrac%7B%5CDelta%20x%7D%7B2%7D%5Cright)%0A%5C%5C%26%3D%5Ccos%20x%0A%5Cend%7Balign%7D$

## 这里用到了重要极限

模块六：余弦函数

$f(x)%3D%5Ccos%20x$

$%5Cbegin%7Balign%7D%0Af'(x)%26%3D%5Clim_%7B%5CDelta%20x%20%5Cto%200%7D%20%5Cfrac%7Bf(x%2B%5CDelta%20x)-f(x)%7D%7B%5CDelta%20x%7D%20%0A%5C%5C%26%3D%5Clim_%7B%5CDelta%20x%20%5Cto%200%7D%20%5Cfrac%7B%5Ccos(x%2B%5CDelta%20x)-%5Ccos%20x%7D%7B%5CDelta%20x%7D%20%0A%5C%5C%26%3D%5Clim_%7B%5CDelta%20x%20%5Cto%200%7D%5Cfrac%7B-2%5Csin%20%5Cleft(x%2B%5Cfrac%7B%5CDelta%20x%7D%7B2%7D%5Cright)%5Csin%5Cfrac%7B%5CDelta%20x%7D%7B2%7D%7D%7B%5CDelta%20x%7D%0A%5C%5C%26%3D%5Clim_%7B%5CDelta%20x%20%5Cto%200%7D%5Cfrac%7B-2%5Csin%20%5Cleft(x%2B%5Cfrac%7B%5CDelta%20x%7D%7B2%7D%5Cright)%5Cfrac%7B%5CDelta%20x%7D%7B2%7D%7D%7B%5CDelta%20x%7D%5C%23%0A%5C%5C%26%3D%5Clim_%7B%5CDelta%20x%5Cto0%7D-%5Csin%5Cleft(x%2B%5Cfrac%7B%5CDelta%20x%7D%7B2%7D%5Cright)%0A%5C%5C%26%3D-%5Csin%20x%0A%5Cend%7Balign%7D$

## 这里用到了重要极限

模块七：正切函数

$f(x)%3D%5Ctan%20x$

$%5Cbegin%7Balign%7D%0Af'(x)%26%3D%5Cleft(%5Cfrac%7B%5Csin%20x%7D%7B%5Ccos%20x%7D%5Cright)'%0A%5C%5C%26%3D%20%5Cfrac%7B%5Ccos%20x%5Ccdot%5Ccos%20x-%5Csin%20x%5Ccdot(-%5Csin%20x)%7D%7B%5Ccos%5E2x%7D%0A%5C%5C%26%3D%5Cfrac%7B1%7D%7B%5Ccos%5E2x%7D%0A%5C%5C%26%3D%5Csec%5E2x%0A%5Cend%7Balign%7D$

模块八：余切函数

$f(x)%3D%5Ccot%20x$

$%5Cbegin%7Balign%7D%0Af'(x)%26%3D%5Cleft(%5Cfrac%7B%5Ccos%20x%7D%7B%5Csin%20x%7D%5Cright)'%0A%5C%5C%26%3D%20%5Cfrac%7B(-%5Csin%20x)%5Ccdot%5Csin%20x-%5Ccos%20x%5Ccdot%5Ccos%20x%7D%7B%5Csin%5E2x%7D%0A%5C%5C%26%3D-%5Cfrac%7B1%7D%7B%5Csin%5E2x%7D%0A%5C%5C%26%3D-%5Ccsc%5E2x%0A%5Cend%7Balign%7D$

模块九：正割函数

$f(x)%3D%5Csec%20x$

$%5Cbegin%7Balign%7D%0Af'(x)%26%3D%5Cleft(%5Cfrac%7B1%7D%7B%5Ccos%20x%7D%5Cright)'%0A%5C%5C%26%3D%20%5Cfrac%7B0-1%5Ccdot(-%5Csin%20x)%7D%7B%5Ccos%5E2x%7D%0A%5C%5C%26%3D%5Cfrac%7B%5Csin%20x%7D%7B%5Ccos%5E2x%7D%0A%5C%5C%26%3D%5Ctan%20x%5Csec%20x%0A%5Cend%7Balign%7D$

模块十：余割函数

$f(x)%3D%5Ccsc%20x$

$%5Cbegin%7Balign%7D%0Af'(x)%26%3D%5Cleft(%5Cfrac%7B1%7D%7B%5Csin%20x%7D%5Cright)'%0A%5C%5C%26%3D%20%5Cfrac%7B0-1%5Ccdot%5Ccos%20x%7D%7B%5Csin%5E2x%7D%0A%5C%5C%26%3D-%5Cfrac%7B%5Ccos%20x%7D%7B%5Csin%5E2x%7D%0A%5C%5C%26%3D-%5Ccot%20x%5Ccsc%20x%0A%5Cend%7Balign%7D$

模块十一：反正弦函数

$f(x)%3D%5Carcsin%20x$

$y%3D%5Carcsin%20x$

$%5Csin%20y%3D%20x$

$%5Ccos%20y%5Ccdot%20%5Cfrac%7Bdy%7D%7Bdx%7D%3D%201$

$%5Cfrac%7Bdy%7D%7Bdx%7D%3D%5Cfrac%7B1%7D%7B%5Ccos%20y%7D$

根据反正弦函数中 $y%5Cin%20%5Cleft%5B-%5Cfrac%7B%5Cpi%7D%7B2%7D%2C%5Cfrac%7B%5Cpi%7D%7B2%7D%5Cright%5D$ 可以得到， $%5Ccos%20y%5Cgeq%200$ ，所以有

$%5Ccos%20y%3D%5Csqrt%7B1-%5Csin%5E2y%7D$

$f'(x)%3D%5Cfrac%7B1%7D%7B%5Csqrt%7B1-x%5E2%7D%7D$

模块十二：反余弦函数

$f(x)%3D%5Carccos%20x$

$y%3D%5Carccos%20x$

$%5Ccos%20y%3D%20x$

$-%5Csin%20y%5Ccdot%5Cfrac%7Bdy%7D%7Bdx%7D%3D1$

$%5Cfrac%7Bdy%7D%7Bdx%7D%3D-%5Cfrac%7B1%7D%7B%5Csin%20y%7D$

根据反余弦函数中 $y%5Cin%20%5Cleft%5B0%2C%5Cpi%5Cright%5D$ 可以得到， $%5Csin%20y%5Cgeq%200$ ，所以有

$%5Csin%20y%3D%5Csqrt%7B1-%5Ccos%5E2y%7D$

$f'(x)%3D-%5Cfrac%7B1%7D%7B%5Csqrt%7B1-x%5E2%7D%7D$

模块十三：反正切函数

$f(x)%3D%5Carctan%20x$

$y%3D%5Carctan%20x$

$%5Ctan%20y%3D%20x$

$%5Cfrac%7B1%7D%7B%5Ccos%5E2y%7D%5Ccdot%5Cfrac%7Bdy%7D%7Bdx%7D%3D1$

$%5Cfrac%7Bdy%7D%7Bdx%7D%3D%5Ccos%5E2y$

根据三角恒等式 $1%2B%5Ctan%5E2y%3D%5Cfrac%7B1%7D%7B%5Ccos%5E2y%7D$ 可以得到，

$f'(x)%3D%5Cfrac%7B1%7D%7B1%2Bx%5E2%7D$

模块十四：反余切函数

$f(x)%3D%5Coperatorname%7Barccot%7D%20x$

$y%3D%5Coperatorname%7Barccot%7D%20x$

$%5Ccot%20y%3D%20x$

$-%5Cfrac%7B1%7D%7B%5Csin%5E2y%7D%5Ccdot%5Cfrac%7Bdy%7D%7Bdx%7D%3D1$

$%5Cfrac%7Bdy%7D%7Bdx%7D%3D-%5Csin%5E2y$

根据三角恒等式 $1%2B%5Ccot%5E2y%3D%5Cfrac%7B1%7D%7B%5Csin%5E2y%7D$ 可以得到，

$f'(x)%3D-%5Cfrac%7B1%7D%7B1%2Bx%5E2%7D$

模块十五：反正割函数

$f(x)%3D%5Coperatorname%7Barcsec%7D%20x$

$y%3D%5Coperatorname%7Barcsec%7D%20x$

$%5Csec%20y%3D%20x$

$%5Cfrac%7B%5Csin%20y%7D%7B%5Ccos%5E2y%7D%5Ccdot%5Cfrac%7Bdy%7D%7Bdx%7D%3D1$

$%5Cfrac%7Bdy%7D%7Bdx%7D%3D%5Cfrac%7B%5Ccos%5E2y%7D%7B%5Csin%20y%7D%3D%5Cfrac%7B1%7D%7Bx%5E2%5Csin%20y%7D$

注意到 $%5Ccos%20y%3D%20%5Cfrac%7B1%7D%7Bx%7D$ ，且 $%5Csin%20y%5Cgeq%200$ ，于是有

$f'(x)%3D%5Cfrac%7B1%7D%7Bx%5E2%5Csqrt%7B1-%5Cfrac%7B1%7D%7Bx%5E2%7D%7D%7D%3D%5Cfrac%7B1%7D%7Bx%5Csqrt%7Bx%5E2-1%7D%7D$

模块十六：反余割函数

$f(x)%3D%5Coperatorname%7Barccsc%7D%20x$

$y%3D%5Coperatorname%7Barccsc%7D%20x$

$%5Ccsc%20y%3D%20x$

$-%5Cfrac%7B%5Ccos%20y%7D%7B%5Csin%5E2y%7D%5Ccdot%5Cfrac%7Bdy%7D%7Bdx%7D%3D1$

$%5Cfrac%7Bdy%7D%7Bdx%7D%3D-%5Cfrac%7B%5Csin%5E2y%7D%7B%5Ccos%20y%7D%3D-%5Cfrac%7B1%7D%7Bx%5E2%5Ccos%20y%7D$

注意到 $%5Csin%20y%3D%20%5Cfrac%7B1%7D%7Bx%7D$ ，且 $%5Ccos%20y%5Cgeq%200$ ，于是有

$f'(x)%3D%5Cfrac%7B1%7D%7Bx%5E2%5Csqrt%7B1-%5Cfrac%7B1%7D%7Bx%5E2%7D%7D%7D%3D%5Cfrac%7B1%7D%7Bx%5Csqrt%7Bx%5E2-1%7D%7D$

对三角函数和反三角函数的概念、定义域、值域等内容陌生的读者可以看一看下面的图。

## 从上到下按讲解顺序

## 这里显示函数名称不一样的原因是GeoGebra没有录入部分函数

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