微积分学习笔记一：极限 导数 微分

1、两个重要极限

[lim_{x ightarrow0}frac{sinleft ( x ight )}{x}=1]

[lim_{n ightarrow 0}left ( 1+frac{1}{n} ight )^{^{n}}=1]

2、计算导数的方法：y=f(x)

（1）求增量的增量: [Delta y=fleft ( x+Delta x ight )-fleft ( x ight )]

(2) 计算比值：

[frac{Delta y}{Delta x}=frac{fleft ( x+Delta x ight )-fleft(x ight )}{Delta x}]

(3)求极限[lim_{Delta x ightarrow0}=frac{Delta y}{Delta x}]

3、初等函数导数

$left ( 1 ight )left (C ight )^{^{'}}=0$

$left ( 2 ight )left (x^{a} ight )^{^{'}}=ax^{a-1}$

$left ( 3 ight )left (log_{a}^{x} ight )^{^{'}}=frac{1}{xlna}left(a>0,a eq 0 ight )$

$left ( 4 ight )left (ln x ight )^{^{'}}=frac{1}{x}$

$left ( 5 ight )left (a^{x} ight )^{^{'}}=a^{x}ln x$

$left ( 6 ight )left (e^{x} ight )^{^{'}}=e^{x}$

$left ( 7 ight )left (sin x ight )^{^{'}}=cos x$

$left ( 8 ight )left (cos x ight )^{^{'}}=-sin x$

$left ( 9 ight )left (tan x ight )^{^{'}}=frac{1}{cos^{2}x}$

$left ( 10 ight )left (frac{1}{tan x} ight )^{^{'}}=-frac{1}{sin^{2}x}$

$left ( 11 ight )left (frac{1}{cos x} ight )^{^{'}}=frac{sin x}{cos^{2}x}$

$left ( 12 ight )left (frac{1}{sin x} ight )^{^{'}}=-frac{cos x}{sin^{2}x}$

$left ( 13 ight )left (arcsin x ight )^{^{'}}=frac{1}{sqrt{1-x^{2}}}left ( -1<x<1 ight )$

$left ( 14 ight )left (arccos x ight )^{^{'}}=-frac{1}{sqrt{1-x^{2}}}left ( -1<x<1 ight )$

$left ( 15 ight )left (arctan x ight )^{^{'}}=frac{1}{1+x^{2}}$

$left ( 16 ight )left (arccot x ight )^{^{'}}=-frac{1}{1+x^{2}}left ( cot x=frac{1}{tan x} ight )$



4、y=f(x)在x处导出存在且f'(x)!=0，当$left | Delta x ight |$很小时有

[f left( x+ Delta x ight ) approx f left(x ight )+f^{^{'}}left(x ight )Delta x ]