微分

一、微分

概念

<img src="https://raw.githubusercontent.com/liangxifeng833/my_program/master/images/math/math-weifen-new-1.png" width="500" />

函数可微条件

• 如果函数可微，那么可导；
• 如果函数可导，那么可微；

<img src="https://raw.githubusercontent.com/liangxifeng833/my_program/master/images/math/math-weifen-new-2.png" width="500" />

求微分

• y=f(x) 的微分也就是函数 y＝f(x)的导数* dx , 所以求微分，也就是求导数；

<img src="https://raw.githubusercontent.com/liangxifeng833/my_program/master/images/math/math-weifen-new-3.png" width="500" />

凑微分，为积分做铺垫

<img src="https://raw.githubusercontent.com/liangxifeng833/my_program/master/images/math/math-weifen-new-4.png" width="500" />

二、微分中值定理

罗尔定理

<img src="https://raw.githubusercontent.com/liangxifeng833/my_program/master/images/math/math-weifen-new-5.png" width="500" />
<img src="https://raw.githubusercontent.com/liangxifeng833/my_program/master/images/math/math-weifen-new-6.png" width="500" />

拉格朗日中值定理 ( 很重要 )

<img src="https://raw.githubusercontent.com/liangxifeng833/my_program/master/images/math/math-weifen-new-7.png" width="500" />
<img src="https://raw.githubusercontent.com/liangxifeng833/my_program/master/images/math/math-weifen-new-8.png" width="500" />
<img src="https://raw.githubusercontent.com/liangxifeng833/my_program/master/images/math/math-weifen-new-9.png" width="500" />

洛必达法则

• 未定式, 0/0型和无穷/无穷型，０*无穷型和无穷－无穷型

<img src="https://raw.githubusercontent.com/liangxifeng833/my_program/master/images/math/math-weifen-new-10.png" width="500" />

<img src="https://raw.githubusercontent.com/liangxifeng833/my_program/master/images/math/math-weifen-new-11.png" width="500" />

• 0/0型 未定式的洛必达法则

<img src="https://raw.githubusercontent.com/liangxifeng833/my_program/master/images/math/math-weifen-new-12.png" width="500" />
<img src="https://raw.githubusercontent.com/liangxifeng833/my_program/master/images/math/math-weifen-new-13.png" width="500" />

• 无穷/无穷型 未定式的洛必达法则

<img src="https://raw.githubusercontent.com/liangxifeng833/my_program/master/images/math/math-weifen-new-14.png" width="500" />
<img src="https://raw.githubusercontent.com/liangxifeng833/my_program/master/images/math/math-weifen-new-15.png" width="500" />
<img src="https://raw.githubusercontent.com/liangxifeng833/my_program/master/images/math/math-weifen-new-16.png" width="500" />

• ０*无穷型和无穷－无穷型　未定式的洛必达法则可以转换为　0/0型　和　无穷/无穷型的未定式的洛必达法则

<img src="https://raw.githubusercontent.com/liangxifeng833/my_program/master/images/math/math-weifen-new-17.png" width="500" />