导数知识点总结（大学数学分析）
在数学中，无论是中学还是大学函数都是非常重要的，任何知识点都能利用函数的观点进行解决，而导数又是函数的精华部分，所以导数将在数学领域起着举足轻重的作用，下面我把大学数学分析的导数部分进行一个总结，共大家参考。
1. 导数的文字定义：略
2. 瞬时速度：略
3. 棒的线密度：![](https://wkrtcs.bdimg.com/rtcs/image?w=184.00&md5sum=ffb8350f007ea96a60705f1472b6542b&sign=3085dbd3c0&rtcs_flag=2&rtcs_ver=3.2&l=webapp&bucketNum=93&ipr={"t":"img","r":"r_3","w":"184.00","h":"41.33","dataType":"gif","c":"word/media/image1.gif"})
，其中表示质量表示增量。
4. 导数的表达式定义：若函数在某点![](https://wkrtcs.bdimg.com/rtcs/image?w=21.33&md5sum=ffb8350f007ea96a60705f1472b6542b&sign=3085dbd3c0&rtcs_flag=2&rtcs_ver=3.2&l=webapp&bucketNum=93&ipr={"t":"img","r":"r_3","w":"21.33","h":"26.67","dataType":"gif","c":"word/media/image4.gif"})
处有导数，则函数在处左右导数相等。
![](https://wkrtcs.bdimg.com/rtcs/image?w=216.00&md5sum=ffb8350f007ea96a60705f1472b6542b&sign=3085dbd3c0&rtcs_flag=2&rtcs_ver=3.2&l=webapp&bucketNum=93&ipr={"t":"img","r":"r_2","w":"216.00","h":"48.00","dataType":"gif","c":"word/media/image5.gif"})
有![](https://wkrtcs.bdimg.com/rtcs/image?w=58.67&md5sum=ffb8350f007ea96a60705f1472b6542b&sign=3085dbd3c0&rtcs_flag=2&rtcs_ver=3.2&l=webapp&bucketNum=93&ipr={"t":"img","r":"r_3","w":"58.67","h":"30.67","dataType":"gif","c":"word/media/image7.gif"})
==
5 曲线的方程：略
6 （导数与函数连续关系）导连关系定理：若![](https://wkrtcs.bdimg.com/rtcs/image?w=34.67&md5sum=ffb8350f007ea96a60705f1472b6542b&sign=3085dbd3c0&rtcs_flag=2&rtcs_ver=3.2&l=webapp&bucketNum=93&ipr={"t":"img","r":"r_3","w":"34.67","h":"22.67","dataType":"gif","c":"word/media/image10.gif"})
在处可导，则必在处连续，反之不然。（光滑处可导，代尖处不可导）
7 导数的计算公式：![](https://wkrtcs.bdimg.com/rtcs/image?w=57.33&md5sum=ffb8350f007ea96a60705f1472b6542b&sign=3085dbd3c0&rtcs_flag=2&rtcs_ver=3.2&l=webapp&bucketNum=93&ipr={"t":"img","r":"r_3","w":"57.33","h":"22.67","dataType":"gif","c":"word/media/image11.gif"})
，是常数，则
![](https://wkrtcs.bdimg.com/rtcs/image?w=66.67&md5sum=ffb8350f007ea96a60705f1472b6542b&sign=3085dbd3c0&rtcs_flag=2&rtcs_ver=3.2&l=webapp&bucketNum=93&ipr={"t":"img","r":"r_3","w":"66.67","h":"29.33","dataType":"gif","c":"word/media/image14.gif"})
，是正整数
![](https://wkrtcs.bdimg.com/rtcs/image?w=68.00&md5sum=ffb8350f007ea96a60705f1472b6542b&sign=3085dbd3c0&rtcs_flag=2&rtcs_ver=3.2&l=webapp&bucketNum=93&ipr={"t":"img","r":"r_3","w":"68.00","h":"29.33","dataType":"gif","c":"word/media/image17.gif"})
，是有理数，
![](https://wkrtcs.bdimg.com/rtcs/image?w=80.00&md5sum=ffb8350f007ea96a60705f1472b6542b&sign=3085dbd3c0&rtcs_flag=2&rtcs_ver=3.2&l=webapp&bucketNum=93&ipr={"t":"img","r":"r_3","w":"80.00","h":"22.67","dataType":"gif","c":"word/media/image20.gif"})
，
![](https://wkrtcs.bdimg.com/rtcs/image?w=81.33&md5sum=ffb8350f007ea96a60705f1472b6542b&sign=3085dbd3c0&rtcs_flag=2&rtcs_ver=3.2&l=webapp&bucketNum=93&ipr={"t":"img","r":"r_3","w":"81.33","h":"22.67","dataType":"gif","c":"word/media/image22.gif"})
，
![](https://wkrtcs.bdimg.com/rtcs/image?w=96.00&md5sum=ffb8350f007ea96a60705f1472b6542b&sign=3085dbd3c0&rtcs_flag=2&rtcs_ver=3.2&l=webapp&bucketNum=93&ipr={"t":"img","r":"r_3","w":"96.00","h":"30.67","dataType":"gif","c":"word/media/image24.gif"})
，，
![](https://wkrtcs.bdimg.com/rtcs/image?w=73.33&md5sum=ffb8350f007ea96a60705f1472b6542b&sign=3085dbd3c0&rtcs_flag=2&rtcs_ver=3.2&l=webapp&bucketNum=93&ipr={"t":"img","r":"r_3","w":"73.33","h":"22.67","dataType":"gif","c":"word/media/image27.gif"})
，
8 函数与反函数的可导关系：原逆定理：设![](https://wkrtcs.bdimg.com/rtcs/image?w=60.00&md5sum=ffb8350f007ea96a60705f1472b6542b&sign=3085dbd3c0&rtcs_flag=2&rtcs_ver=3.2&l=webapp&bucketNum=93&ipr={"t":"img","r":"r_3","w":"60.00","h":"22.67","dataType":"gif","c":"word/media/image29.gif"})
在包含的区间上连续严格单调，是在处可导，且，那么它的反函数在可导，且
9 幂指函数;形如![](https://wkrtcs.bdimg.com/rtcs/image?w=104.00&md5sum=ffb8350f007ea96a60705f1472b6542b&sign=3085dbd3c0&rtcs_flag=2&rtcs_ver=3.2&l=webapp&bucketNum=93&ipr={"t":"img","r":"r_3","w":"104.00","h":"33.33","dataType":"gif","c":"word/media/image35.gif"})
，则
10 高阶导数：形如![](https://wkrtcs.bdimg.com/rtcs/image?w=37.33&md5sum=ffb8350f007ea96a60705f1472b6542b&sign=3085dbd3c0&rtcs_flag=2&rtcs_ver=3.2&l=webapp&bucketNum=93&ipr={"t":"img","r":"r_3","w":"37.33","h":"22.67","dataType":"gif","c":"word/media/image37.gif"})
一阶导数，二阶导数，三阶导数…，阶导数
11 加减法导数定理：![](https://wkrtcs.bdimg.com/rtcs/image?w=160.00&md5sum=ffb8350f007ea96a60705f1472b6542b&sign=3085dbd3c0&rtcs_flag=2&rtcs_ver=3.2&l=webapp&bucketNum=93&ipr={"t":"img","r":"r_5","w":"160.00","h":"33.33","dataType":"gif","c":"word/media/image41.gif"})
，
12 导数乘法定理（莱布尼兹定理）：设![](https://wkrtcs.bdimg.com/rtcs/image?w=16.00&md5sum=ffb8350f007ea96a60705f1472b6542b&sign=3085dbd3c0&rtcs_flag=2&rtcs_ver=3.2&l=webapp&bucketNum=93&ipr={"t":"img","r":"r_5","w":"16.00","h":"21.33","dataType":"gif","c":"word/media/image43.gif"})
，有阶导数，则也有阶导数
13 函数极值的定义：![](https://wkrtcs.bdimg.com/rtcs/image?w=88.00&md5sum=ffb8350f007ea96a60705f1472b6542b&sign=3085dbd3c0&rtcs_flag=2&rtcs_ver=3.2&l=webapp&bucketNum=93&ipr={"t":"img","r":"r_5","w":"88.00","h":"22.67","dataType":"gif","c":"word/media/image47.gif"})
,，，
若![](https://wkrtcs.bdimg.com/rtcs/image?w=86.67&md5sum=ffb8350f007ea96a60705f1472b6542b&sign=3085dbd3c0&rtcs_flag=2&rtcs_ver=3.2&l=webapp&bucketNum=93&ipr={"t":"img","r":"r_5","w":"86.67","h":"26.67","dataType":"gif","c":"word/media/image51.gif"})
，则称是极小值点，是极小值点。
![](https://wkrtcs.bdimg.com/rtcs/image?w=86.67&md5sum=ffb8350f007ea96a60705f1472b6542b&sign=3085dbd3c0&rtcs_flag=2&rtcs_ver=3.2&l=webapp&bucketNum=93&ipr={"t":"img","r":"r_5","w":"86.67","h":"26.67","dataType":"gif","c":"word/media/image53.gif"})
则称是极大值点，是极大值点。
14 极值点定理：设![](https://wkrtcs.bdimg.com/rtcs/image?w=21.33&md5sum=ffb8350f007ea96a60705f1472b6542b&sign=3085dbd3c0&rtcs_flag=2&rtcs_ver=3.2&l=webapp&bucketNum=93&ipr={"t":"img","r":"r_5","w":"21.33","h":"26.67","dataType":"gif","c":"word/media/image4.gif"})
是的极值点，如果在处可导，则
15 柔勒定理：设![](https://wkrtcs.bdimg.com/rtcs/image?w=34.67&md5sum=ffb8350f007ea96a60705f1472b6542b&sign=3085dbd3c0&rtcs_flag=2&rtcs_ver=3.2&l=webapp&bucketNum=93&ipr={"t":"img","r":"r_5","w":"34.67","h":"22.67","dataType":"gif","c":"word/media/image10.gif"})
满足
①在![](https://wkrtcs.bdimg.com/rtcs/image?w=34.67&md5sum=ffb8350f007ea96a60705f1472b6542b&sign=3085dbd3c0&rtcs_flag=2&rtcs_ver=3.2&l=webapp&bucketNum=93&ipr={"t":"img","r":"r_5","w":"34.67","h":"22.67","dataType":"gif","c":"word/media/image55.gif"})
上连续
②在![](https://wkrtcs.bdimg.com/rtcs/image?w=36.00&md5sum=ffb8350f007ea96a60705f1472b6542b&sign=3085dbd3c0&rtcs_flag=2&rtcs_ver=3.2&l=webapp&bucketNum=93&ipr={"t":"img","r":"r_5","w":"36.00","h":"22.67","dataType":"gif","c":"word/media/image56.gif"})
上可导
③![](https://wkrtcs.bdimg.com/rtcs/image?w=78.67&md5sum=ffb8350f007ea96a60705f1472b6542b&sign=3085dbd3c0&rtcs_flag=2&rtcs_ver=3.2&l=webapp&bucketNum=93&ipr={"t":"img","r":"r_6","w":"78.67","h":"22.67","dataType":"gif","c":"word/media/image57.gif"})
则![](https://wkrtcs.bdimg.com/rtcs/image?w=70.67&md5sum=ffb8350f007ea96a60705f1472b6542b&sign=3085dbd3c0&rtcs_flag=2&rtcs_ver=3.2&l=webapp&bucketNum=93&ipr={"t":"img","r":"r_7","w":"70.67","h":"22.67","dataType":"gif","c":"word/media/image58.gif"})
，则
![](https://wkrtcs.bdimg.com/rtcs/image?w=96.00&md5sum=ffb8350f007ea96a60705f1472b6542b&sign=3085dbd3c0&rtcs_flag=2&rtcs_ver=3.2&l=webapp&bucketNum=93&ipr={"t":"img","r":"r_3","w":"96.00","h":"22.67","dataType":"gif","c":"word/media/image60.gif"})
，
未完待续
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