微积分学习笔记一：极限 导数 微分

1、两个重要极限

\[\lim_{x\rightarrow0}\frac{sin\left ( x \right )}{x}=1\]

\[\lim_{n\rightarrow 0}\left ( 1+\frac{1}{n} \right )^{^{n}}=1\]

2、计算导数的方法：y=f(x)

（1）求增量的增量: \[\Delta y=f\left ( x+\Delta x \right )-f\left ( x \right )\]

(2) 计算比值：

\[\frac{\Delta y}{\Delta x}=\frac{f\left ( x+\Delta x \right )-f\left(x \right )}{\Delta x}\]

(3)求极限\[\lim_{\Delta x\rightarrow0}=\frac{\Delta y}{\Delta x}\]

3、初等函数导数

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

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

$\left ( 3 \right )\left (log_{a}^{x} \right )^{^{'}}=\frac{1}{xlna}\left(a>0,a\neq 0 \right )$

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

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

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

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

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

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

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

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

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

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

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

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

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



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

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