§2.3 反函数的导数，复合函数的求导法则
一、反函数的导数
设![](https://wkrtcs.bdimg.com/rtcs/image?w=60&md5sum=63c4e8d1ce4cffed1ca55098512a9aa3&sign=80db979387&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=148&ipr={"t":"img","w":"60","h":"21","dataType":"png","c":"word/media/image1_1.png","imgOriW":1440,"imgOriH":512})
是直接函数，是它的反函数，假定在内单调、可导，而且，则反函数在间内也是单调、可导的，而且
![](https://wkrtcs.bdimg.com/rtcs/image?w=92&md5sum=63c4e8d1ce4cffed1ca55098512a9aa3&sign=80db979387&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=148&ipr={"t":"img","w":"92","h":"44","dataType":"png","c":"word/media/image8_1.png","imgOriW":1913,"imgOriH":915})
(1)
证明：![](https://wkrtcs.bdimg.com/rtcs/image?w=67&md5sum=63c4e8d1ce4cffed1ca55098512a9aa3&sign=80db979387&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=148&ipr={"t":"img","w":"67","h":"27","dataType":"png","c":"word/media/image9_1.png","imgOriW":1600,"imgOriH":640})
，给以增量
由 ![](https://wkrtcs.bdimg.com/rtcs/image?w=61&md5sum=63c4e8d1ce4cffed1ca55098512a9aa3&sign=80db979387&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=148&ipr={"t":"img","w":"61","h":"21","dataType":"png","c":"word/media/image13_1.png","imgOriW":1472,"imgOriH":512})
在 上的单调性可知
![](https://wkrtcs.bdimg.com/rtcs/image?w=175&md5sum=63c4e8d1ce4cffed1ca55098512a9aa3&sign=80db979387&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=148&ipr={"t":"img","w":"175","h":"21","dataType":"png","c":"word/media/image15_1.png","imgOriW":3771,"imgOriH":461})
于是 ![](https://wkrtcs.bdimg.com/rtcs/image?w=59&md5sum=63c4e8d1ce4cffed1ca55098512a9aa3&sign=80db979387&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=148&ipr={"t":"img","w":"59","h":"61","dataType":"png","c":"word/media/image16_1.png","imgOriW":1326,"imgOriH":1326})
因直接函数![](https://wkrtcs.bdimg.com/rtcs/image?w=60&md5sum=63c4e8d1ce4cffed1ca55098512a9aa3&sign=80db979387&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=148&ipr={"t":"img","w":"60","h":"21","dataType":"png","c":"word/media/image17_1.png","imgOriW":1440,"imgOriH":512})
在上单调、可导，故它是连续的，且反函数在上也是连续的，当时，必有
![](https://wkrtcs.bdimg.com/rtcs/image?w=173&md5sum=63c4e8d1ce4cffed1ca55098512a9aa3&sign=80db979387&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=148&ipr={"t":"img","w":"173","h":"65","dataType":"png","c":"word/media/image23_1.png","imgOriW":2232,"imgOriH":841})
即：![](https://wkrtcs.bdimg.com/rtcs/image?w=92&md5sum=63c4e8d1ce4cffed1ca55098512a9aa3&sign=80db979387&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=148&ipr={"t":"img","w":"92","h":"44","dataType":"png","c":"word/media/image24_1.png","imgOriW":1913,"imgOriH":915})
【例1】试证明下列基本导数公式
![](https://wkrtcs.bdimg.com/rtcs/image?w=208&md5sum=63c4e8d1ce4cffed1ca55098512a9aa3&sign=80db979387&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=148&ipr={"t":"img","w":"208","h":"125","dataType":"png","c":"word/media/image25_1.png","imgOriW":1598,"imgOriH":1229})
证1、设![](https://wkrtcs.bdimg.com/rtcs/image?w=61&md5sum=63c4e8d1ce4cffed1ca55098512a9aa3&sign=80db979387&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=148&ipr={"t":"img","w":"61","h":"21","dataType":"png","c":"word/media/image26_1.png","imgOriW":1472,"imgOriH":512})
为直接函数，是它的反函数
函数 ![](https://wkrtcs.bdimg.com/rtcs/image?w=61&md5sum=63c4e8d1ce4cffed1ca55098512a9aa3&sign=80db979387&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=148&ipr={"t":"img","w":"61","h":"21","dataType":"png","c":"word/media/image28_1.png","imgOriW":1472,"imgOriH":512})
在 上单调、可导，且
因此，在 ![](https://wkrtcs.bdimg.com/rtcs/image?w=87&md5sum=63c4e8d1ce4cffed1ca55098512a9aa3&sign=80db979387&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=148&ipr={"t":"img","w":"87","h":"24","dataType":"png","c":"word/media/image31_1.png","imgOriW":2080,"imgOriH":576})
上， 有
![](https://wkrtcs.bdimg.com/rtcs/image?w=119&md5sum=63c4e8d1ce4cffed1ca55098512a9aa3&sign=80db979387&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=148&ipr={"t":"img","w":"119","h":"44","dataType":"png","c":"word/media/image32_1.png","imgOriW":2195,"imgOriH":814})
注意到，当![](https://wkrtcs.bdimg.com/rtcs/image?w=80&md5sum=63c4e8d1ce4cffed1ca55098512a9aa3&sign=80db979387&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=148&ipr={"t":"img","w":"80","h":"41","dataType":"png","c":"word/media/image33_1.png","imgOriW":1920,"imgOriH":992})
时，，
因此， ![](https://wkrtcs.bdimg.com/rtcs/image?w=136&md5sum=63c4e8d1ce4cffed1ca55098512a9aa3&sign=80db979387&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=148&ipr={"t":"img","w":"136","h":"45","dataType":"png","c":"word/media/image36_1.png","imgOriW":2329,"imgOriH":776})
证2 设![](https://wkrtcs.bdimg.com/rtcs/image?w=61&md5sum=63c4e8d1ce4cffed1ca55098512a9aa3&sign=80db979387&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=148&ipr={"t":"img","w":"61","h":"23","dataType":"png","c":"word/media/image37_1.png","imgOriW":1472,"imgOriH":544})
，
则![](https://wkrtcs.bdimg.com/rtcs/image?w=89&md5sum=63c4e8d1ce4cffed1ca55098512a9aa3&sign=80db979387&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=148&ipr={"t":"img","w":"89","h":"23","dataType":"png","c":"word/media/image39_1.png","imgOriW":2144,"imgOriH":544})
，
![](https://wkrtcs.bdimg.com/rtcs/image?w=49&md5sum=63c4e8d1ce4cffed1ca55098512a9aa3&sign=80db979387&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=148&ipr={"t":"img","w":"49","h":"19","dataType":"png","c":"word/media/image41_1.png","imgOriW":1184,"imgOriH":448})
在 上单调、可导且
故 ![](https://wkrtcs.bdimg.com/rtcs/image?w=296&md5sum=63c4e8d1ce4cffed1ca55098512a9aa3&sign=80db979387&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=148&ipr={"t":"img","w":"296","h":"44","dataType":"png","c":"word/media/image44_1.png","imgOriW":3567,"imgOriH":530})
证3 ![](https://wkrtcs.bdimg.com/rtcs/image?w=224&md5sum=63c4e8d1ce4cffed1ca55098512a9aa3&sign=80db979387&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=148&ipr={"t":"img","w":"224","h":"45","dataType":"png","c":"word/media/image45_1.png","imgOriW":3037,"imgOriH":615})
类似地，我们可以证明下列导数公式：
![](https://wkrtcs.bdimg.com/rtcs/image?w=189&md5sum=63c4e8d1ce4cffed1ca55098512a9aa3&sign=80db979387&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=148&ipr={"t":"img","w":"189","h":"134","dataType":"png","c":"word/media/image46_1.png","imgOriW":1523,"imgOriH":1287})
二、复合函数的求导法则
如果![](https://wkrtcs.bdimg.com/rtcs/image?w=60&md5sum=63c4e8d1ce4cffed1ca55098512a9aa3&sign=80db979387&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=148&ipr={"t":"img","w":"60","h":"21","dataType":"png","c":"word/media/image47_1.png","imgOriW":1440,"imgOriH":512})
在点可导，而在点可导，则复合函数在点可导，且导数为
![](https://wkrtcs.bdimg.com/rtcs/image?w=153&md5sum=63c4e8d1ce4cffed1ca55098512a9aa3&sign=80db979387&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=148&ipr={"t":"img","w":"153","h":"49","dataType":"png","c":"word/media/image52_1.png","imgOriW":2387,"imgOriH":768})
证明：因![](https://wkrtcs.bdimg.com/rtcs/image?w=111&md5sum=63c4e8d1ce4cffed1ca55098512a9aa3&sign=80db979387&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=148&ipr={"t":"img","w":"111","h":"41","dataType":"png","c":"word/media/image53_1.png","imgOriW":2175,"imgOriH":813})
，由极限与无穷小的关系，有
![](https://wkrtcs.bdimg.com/rtcs/image?w=281&md5sum=63c4e8d1ce4cffed1ca55098512a9aa3&sign=80db979387&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=148&ipr={"t":"img","w":"281","h":"24","dataType":"png","c":"word/media/image54_1.png","imgOriW":4619,"imgOriH":394})
用![](https://wkrtcs.bdimg.com/rtcs/image?w=47&md5sum=63c4e8d1ce4cffed1ca55098512a9aa3&sign=80db979387&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=148&ipr={"t":"img","w":"47","h":"19","dataType":"png","c":"word/media/image55_1.png","imgOriW":1120,"imgOriH":448})
去除上式两边得：
![](https://wkrtcs.bdimg.com/rtcs/image?w=169&md5sum=63c4e8d1ce4cffed1ca55098512a9aa3&sign=80db979387&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=148&ipr={"t":"img","w":"169","h":"41","dataType":"png","c":"word/media/image56_1.png","imgOriW":2734,"imgOriH":667})
由![](https://wkrtcs.bdimg.com/rtcs/image?w=60&md5sum=63c4e8d1ce4cffed1ca55098512a9aa3&sign=80db979387&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=148&ipr={"t":"img","w":"60","h":"21","dataType":"png","c":"word/media/image57_1.png","imgOriW":1440,"imgOriH":512})
在的可导性有：
![](https://wkrtcs.bdimg.com/rtcs/image?w=128&md5sum=63c4e8d1ce4cffed1ca55098512a9aa3&sign=80db979387&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=148&ipr={"t":"img","w":"128","h":"19","dataType":"png","c":"word/media/image59_1.png","imgOriW":3072,"imgOriH":448})
，
![](https://wkrtcs.bdimg.com/rtcs/image?w=229&md5sum=63c4e8d1ce4cffed1ca55098512a9aa3&sign=80db979387&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=148&ipr={"t":"img","w":"229","h":"41","dataType":"png","c":"word/media/image61_1.png","imgOriW":3212,"imgOriH":579})
![](https://wkrtcs.bdimg.com/rtcs/image?w=217&md5sum=63c4e8d1ce4cffed1ca55098512a9aa3&sign=80db979387&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=148&ipr={"t":"img","w":"217","h":"41","dataType":"png","c":"word/media/image62_1.png","imgOriW":3122,"imgOriH":594})
![](https://wkrtcs.bdimg.com/rtcs/image?w=107&md5sum=63c4e8d1ce4cffed1ca55098512a9aa3&sign=80db979387&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=148&ipr={"t":"img","w":"107","h":"24","dataType":"png","c":"word/media/image63_1.png","imgOriW":2560,"imgOriH":576})
即![](https://wkrtcs.bdimg.com/rtcs/image?w=153&md5sum=63c4e8d1ce4cffed1ca55098512a9aa3&sign=80db979387&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=148&ipr={"t":"img","w":"153","h":"49","dataType":"png","c":"word/media/image64_1.png","imgOriW":2387,"imgOriH":768})
上述复合函数的求导法则可作更一般的叙述：
若![](https://wkrtcs.bdimg.com/rtcs/image?w=73&md5sum=63c4e8d1ce4cffed1ca55098512a9aa3&sign=80db979387&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=148&ipr={"t":"img","w":"73","h":"25","dataType":"png","c":"word/media/image65_1.png","imgOriW":1760,"imgOriH":608})
在开区间可导，在开区间可导，且时，对应的 ，则复合函数在内可导，且
![](https://wkrtcs.bdimg.com/rtcs/image?w=88&md5sum=63c4e8d1ce4cffed1ca55098512a9aa3&sign=80db979387&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=148&ipr={"t":"img","w":"88","h":"41","dataType":"png","c":"word/media/image73_1.png","imgOriW":2112,"imgOriH":992})
(2)
复合函数求导法则是一个非常重要的法则，特给出如下注记：
![](https://wkrtcs.bdimg.com/rtcs/image?w=523&md5sum=63c4e8d1ce4cffed1ca55098512a9aa3&sign=80db979387&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=148&ipr={"t":"img","w":"523","h":"210","dataType":"png","c":"word/media/image74.png","imgOriW":524,"imgOriH":210})
弄懂了锁链规则的实质之后，不难给出复合更多层函数的求导公式。
【例2】![](https://wkrtcs.bdimg.com/rtcs/image?w=119&md5sum=63c4e8d1ce4cffed1ca55098512a9aa3&sign=80db979387&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=148&ipr={"t":"img","w":"119","h":"21","dataType":"png","c":"word/media/image75_1.png","imgOriW":2848,"imgOriH":512})
，求
引入中间变量， 设 ![](https://wkrtcs.bdimg.com/rtcs/image?w=72&md5sum=63c4e8d1ce4cffed1ca55098512a9aa3&sign=80db979387&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=148&ipr={"t":"img","w":"72","h":"25","dataType":"png","c":"word/media/image77_1.png","imgOriW":1728,"imgOriH":608})
，，于是
变量关系是 ![](https://wkrtcs.bdimg.com/rtcs/image?w=104&md5sum=63c4e8d1ce4cffed1ca55098512a9aa3&sign=80db979387&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=148&ipr={"t":"img","w":"104","h":"20","dataType":"png","c":"word/media/image80_1.png","imgOriW":2496,"imgOriH":480})
，由锁链规则有：
![](https://wkrtcs.bdimg.com/rtcs/image?w=137&md5sum=63c4e8d1ce4cffed1ca55098512a9aa3&sign=80db979387&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=148&ipr={"t":"img","w":"137","h":"49","dataType":"png","c":"word/media/image81_1.png","imgOriW":2251,"imgOriH":809})
(2)、用锁链规则求导的关键
引入中间变量，将复合函数分解成基本初等函数。还应注意：求导完成后，应将引入的中间变量代换成原自变量。
【例3】求![](https://wkrtcs.bdimg.com/rtcs/image?w=84&md5sum=63c4e8d1ce4cffed1ca55098512a9aa3&sign=80db979387&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=148&ipr={"t":"img","w":"84","h":"25","dataType":"png","c":"word/media/image82_1.png","imgOriW":2016,"imgOriH":608})
的导数。
解：设 ![](https://wkrtcs.bdimg.com/rtcs/image?w=57&md5sum=63c4e8d1ce4cffed1ca55098512a9aa3&sign=80db979387&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=148&ipr={"t":"img","w":"57","h":"19","dataType":"png","c":"word/media/image84_1.png","imgOriW":1376,"imgOriH":448})
，则，，由锁链规则有：
![](https://wkrtcs.bdimg.com/rtcs/image?w=408&md5sum=63c4e8d1ce4cffed1ca55098512a9aa3&sign=80db979387&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=148&ipr={"t":"img","w":"408","h":"49","dataType":"png","c":"word/media/image87_1.png","imgOriW":3995,"imgOriH":483})
【例4】 设 ![](https://wkrtcs.bdimg.com/rtcs/image?w=88&md5sum=63c4e8d1ce4cffed1ca55098512a9aa3&sign=80db979387&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=148&ipr={"t":"img","w":"88","h":"49","dataType":"png","c":"word/media/image88_1.png","imgOriW":1772,"imgOriH":993})
，求。
![](https://wkrtcs.bdimg.com/rtcs/image?w=429&md5sum=63c4e8d1ce4cffed1ca55098512a9aa3&sign=80db979387&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=148&ipr={"t":"img","w":"429","h":"169","dataType":"png","c":"word/media/image90.png","imgOriW":430,"imgOriH":169})
由锁链规则有 ![](https://wkrtcs.bdimg.com/rtcs/image?w=120&md5sum=63c4e8d1ce4cffed1ca55098512a9aa3&sign=80db979387&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=148&ipr={"t":"img","w":"120","h":"44","dataType":"png","c":"word/media/image91_1.png","imgOriW":2190,"imgOriH":808})
![](https://wkrtcs.bdimg.com/rtcs/image?w=112&md5sum=63c4e8d1ce4cffed1ca55098512a9aa3&sign=80db979387&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=148&ipr={"t":"img","w":"112","h":"42","dataType":"png","c":"word/media/image92_1.png","imgOriW":2044,"imgOriH":856})
(基本初等函数求导)
![](https://wkrtcs.bdimg.com/rtcs/image?w=120&md5sum=63c4e8d1ce4cffed1ca55098512a9aa3&sign=80db979387&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=148&ipr={"t":"img","w":"120","h":"61","dataType":"png","c":"word/media/image93_1.png","imgOriW":1893,"imgOriH":967})
( 消中间变量)
![](https://wkrtcs.bdimg.com/rtcs/image?w=51&md5sum=63c4e8d1ce4cffed1ca55098512a9aa3&sign=80db979387&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=148&ipr={"t":"img","w":"51","h":"41","dataType":"png","c":"word/media/image94_1.png","imgOriW":1216,"imgOriH":992})
由上例，不难发现复合函数求导窍门
中间变量在求导过程中，只是起过渡作用，熟练之后，可不必引入，仅需“心中有链”。
![](https://wkrtcs.bdimg.com/rtcs/image?w=496&md5sum=63c4e8d1ce4cffed1ca55098512a9aa3&sign=80db979387&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=148&ipr={"t":"img","w":"496","h":"71","dataType":"png","c":"word/media/image95.png","imgOriW":497,"imgOriH":71})
然后，对函数所有中间变量求导，直至求到自变量为止，最后诸导数相乘。
请看下面的演示过程：
![](https://wkrtcs.bdimg.com/rtcs/image?w=360&md5sum=63c4e8d1ce4cffed1ca55098512a9aa3&sign=80db979387&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=148&ipr={"t":"img","w":"360","h":"61","dataType":"png","c":"word/media/image96_1.png","imgOriW":3171,"imgOriH":608})
![](https://wkrtcs.bdimg.com/rtcs/image?w=352&md5sum=63c4e8d1ce4cffed1ca55098512a9aa3&sign=80db979387&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=148&ipr={"t":"img","w":"352","h":"61","dataType":"png","c":"word/media/image97_1.png","imgOriW":3144,"imgOriH":613})
【例5】证明幂函数的导数公式 ![](https://wkrtcs.bdimg.com/rtcs/image?w=128&md5sum=63c4e8d1ce4cffed1ca55098512a9aa3&sign=80db979387&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=148&ipr={"t":"img","w":"128","h":"32","dataType":"png","c":"word/media/image98_1.png","imgOriW":2272,"imgOriH":576})
，(为实数)。
证明：设![](https://wkrtcs.bdimg.com/rtcs/image?w=125&md5sum=63c4e8d1ce4cffed1ca55098512a9aa3&sign=80db979387&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=148&ipr={"t":"img","w":"125","h":"31","dataType":"png","c":"word/media/image100_1.png","imgOriW":2668,"imgOriH":653})
![](https://wkrtcs.bdimg.com/rtcs/image?w=280&md5sum=63c4e8d1ce4cffed1ca55098512a9aa3&sign=80db979387&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=148&ipr={"t":"img","w":"280","h":"49","dataType":"png","c":"word/media/image101_1.png","imgOriW":3488,"imgOriH":538})
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