导数的计算

1．下列求导运算正确的是

A．word/media/image1.gif B．word/media/image2.gif

C．word/media/image3.gif D．word/media/image4.gif

2．已知函数word/media/image5.gif在点P处的导数值为3，则P点的坐标为

A．word/media/image6.gif B．word/media/image7.gif

C．word/media/image6.gif或word/media/image8.gif D．word/media/image7.gif或word/media/image9.gif

3．已知函数word/media/image10.gif的导函数为word/media/image11.gif，且满足word/media/image12.gif，则word/media/image13.gif等于

A．word/media/image14.gif B．word/media/image15.gif C．1 D．e

4．曲线word/media/image16.gif在点word/media/image17.gif处的切线与坐标轴所围成的三角形的面积为

A．word/media/image18.gif B．word/media/image19.gif C．word/media/image20.gif D．word/media/image21.gif

5．已知函数word/media/image22.gif，则word/media/image23.gif

A．word/media/image24.gif B．word/media/image25.gif C．1 D．0

6．函数word/media/image26.gif的导函数为 .

7．设函数word/media/image27.gif，若word/media/image28.gif，则word/media/image29.gif= .

8．求下列函数的导数：

（1）word/media/image30.gif； （2）word/media/image31.gif；

（3）word/media/image32.gif； （4）word/media/image33.gif.

9．已知word/media/image34.gif，则word/media/image35.gif等于

A．word/media/image36.gif B．word/media/image37.gif C．word/media/image38.gif D．word/media/image39.gif

10．已知word/media/image40.gif为自然对数的底数，曲线word/media/image41.gif在点word/media/image42.gif处的切线与直线word/media/image43.gif平行，则实数word/media/image44.gif

A．word/media/image45.gif B．word/media/image46.gif C．word/media/image47.gif D．word/media/image48.gif

11．若word/media/image49.gif，则word/media/image50.gif的解集为

A．word/media/image51.gif B．word/media/image52.gif C．word/media/image53.gif D．word/media/image54.gif

12．已知函数word/media/image55.gif的导函数为word/media/image56.gif，且满足word/media/image57.gif，则word/media/image58.gif .

13．设曲线word/media/image59.gif在点word/media/image60.gif处的切线与word/media/image61.gif轴的交点的横坐标为word/media/image62.gif令word/media/image63.gif则word/media/image64.gif的值为 ．

14．设函数word/media/image65.gif，曲线word/media/image66.gif在点word/media/image67.gif处的切线方程为word/media/image68.gif.

（1）求word/media/image10.gif的解析式；

（2）证明：曲线word/media/image66.gif上任一点处的切线与直线word/media/image69.gif和直线word/media/image70.gif所围成的三角形的面积为定值，并求此定值.

15．设直线l1，l2分别是函数word/media/image71.gif图象上点P1，P2处的切线，l1与l2垂直相交于点P，且l1，l2分别与y轴相交于点A，B，则word/media/image72.gifPAB的面积的取值范围是

A．word/media/image73.gif B．word/media/image74.gif C．word/media/image75.gif D．word/media/image76.gif

16.已知word/media/image10.gif为偶函数，当word/media/image77.gif时，word/media/image78.gif，则曲线word/media/image66.gif在点word/media/image79.gif处的切线方程是_________.

17．已知函数word/media/image80.gif，其中a为实数，word/media/image81.gif为word/media/image82.gif的导函数,若word/media/image83.gif，则a的值为 ．

18．已知曲线word/media/image84.gif在点word/media/image9.gif处的切线与曲线word/media/image85.gif相切，则word/media/image86.gif .

参考答案

1．B 【解析】因为word/media/image87.gif，所以A项应为word/media/image88.gif；由word/media/image89.gif知B项正确；由word/media/image90.gif可知C项错误；D项中，word/media/image91.gif，所以D项是错误的，综上所述，正确选项为B.

2．D 【解析】由word/media/image92.gif得word/media/image93.gif，令word/media/image94.gif，则word/media/image95.gif，故P点的坐标为word/media/image7.gif或word/media/image9.gif.

3．B 【解析】∵函数word/media/image10.gif的导函数为word/media/image11.gif，且满足word/media/image96.gif，

∴word/media/image97.gif，把word/media/image98.gif代入word/media/image11.gif可得word/media/image99.gif，解得word/media/image100.gif.

4．A 【解析】因为word/media/image101.gif，所以切线的斜率为word/media/image102.gif，切线方程为word/media/image103.gif，令word/media/image104.gif得word/media/image105.gif；令word/media/image106.gif得word/media/image107.gif，故围成的三角形的面积为word/media/image108.gif，故选A.

5．C 【解析】由题意得，word/media/image109.gif，令word/media/image110.gif，则word/media/image111.gif，解得word/media/image112.gif，即word/media/image113.gif，

所以word/media/image114.gif，故选C．

6．word/media/image115.gif 【解析】word/media/image116.gif，则word/media/image117.gif.

7．word/media/image40.gif 【解析】由题意得word/media/image118.gif，又word/media/image119.gif，解得word/media/image120.gif．

8．【解析】（1）因为word/media/image121.gif，所以word/media/image122.gif.

（2）word/media/image123.gif.

（3）word/media/image124.gifword/media/image125.gif

word/media/image126.gif.

（4）∵word/media/image127.gif，∴word/media/image128.gif.

9．C 【解析】令word/media/image129.gif，则word/media/image130.gif，word/media/image131.gif，因此word/media/image132.gif，则根据求导公式有word/media/image133.gif，所以word/media/image134.gif.故选C.

10．B 【解析】word/media/image41.gif的导数为word/media/image135.gif，可得曲线word/media/image41.gif在点word/media/image42.gif处的切线斜率为word/media/image136.gif，由切线与直线word/media/image43.gif平行，可得word/media/image137.gif，解得word/media/image138.gif．故选B．

11．C 【解析】要使函数有意义，则word/media/image139.gif，∵word/media/image49.gif，∴word/media/image140.gif，若word/media/image50.gif，则word/media/image141.gif，即word/media/image142.gif，解得word/media/image143.gif或word/media/image144.gif（舍去），故不等式word/media/image50.gif的解集为word/media/image53.gif，故选C.

12．word/media/image145.gif 【解析】∵word/media/image57.gif，∴word/media/image146.gif，令word/media/image98.gif，则word/media/image147.gif，∴word/media/image148.gif；令word/media/image69.gif，则word/media/image149.gif，∴word/media/image150.gif，

∴word/media/image151.gif.

13．word/media/image152.gif 【解析】导函数word/media/image153.gif，切线斜率word/media/image154.gif，所以切线方程为word/media/image155.gif，可求得切线与横轴的交点为word/media/image156.gif，则word/media/image157.gif，所以有word/media/image64.gifword/media/image158.gif.

14．【解析】（1）由word/media/image68.gif得word/media/image159.gif.

当word/media/image160.gif时，word/media/image161.gif，则word/media/image162.gif①.

又word/media/image163.gif，则word/media/image164.gif②.

由①②得word/media/image165.gif，解得word/media/image166.gif.

故word/media/image167.gif.

（2）设word/media/image168.gif为曲线上任一点，由word/media/image169.gif，知曲线在点word/media/image168.gif处的切线方程为word/media/image170.gif，即word/media/image171.gif.

令word/media/image69.gif得word/media/image172.gif，从而得切线与直线word/media/image69.gif的交点坐标为word/media/image173.gif；

令word/media/image70.gif得word/media/image174.gif，从而得切线与直线word/media/image70.gif的交点坐标为word/media/image175.gif，

所以点word/media/image168.gif处的切线与直线word/media/image69.gif，word/media/image70.gif所围成的三角形的面积为word/media/image176.gif.

故曲线word/media/image66.gif上任一点处的切线与直线word/media/image69.gif和直线word/media/image70.gif所围成的三角形的面积为定值，此定值为6.

15．A 【解析】设word/media/image177.gif（不妨设word/media/image178.gif），则由导数的几何意义易得切线word/media/image179.gif的斜率分别为word/media/image180.gif由已知得word/media/image181.gif切线word/media/image182.gif的方程为word/media/image183.gif，切线word/media/image184.gif的方程为word/media/image185.gif，即word/media/image186.gif.分别令word/media/image187.gif得word/media/image188.gifword/media/image182.gif与word/media/image184.gif的交点为word/media/image189.gif

word/media/image190.gif，故选A.

16．word/media/image191.gif 【解析】当word/media/image192.gif时，word/media/image193.gif，则word/media/image194.gif．又因为word/media/image10.gif为偶函数，所以word/media/image195.gif，所以word/media/image196.gif，则word/media/image197.gif，所以切线方程为word/media/image198.gif，即word/media/image191.gif．

17．3 【解析】因为word/media/image199.gif，所以word/media/image200.gif.

18．8 【解析】因为word/media/image201.gif，所以word/media/image202.gif，则曲线word/media/image84.gif在点word/media/image9.gif处的切线方程为word/media/image203.gif，即word/media/image204.gif.又切线与曲线word/media/image85.gif相切，当word/media/image205.gif时，word/media/image206.gif，显然与word/media/image204.gif平行，故word/media/image207.gif，由word/media/image208.gif，得word/media/image209.gif，则word/media/image210.gif，解得word/media/image211.gif