1、复合函数的导数复合函数的导数一、复习与引入：一、复习与引入：1.函数的导数的定义与几何意义函数的导数的定义与几何意义.2.常见函数的导数公式常见函数的导数公式.3.导数的四则运算法则导数的四则运算法则.4.例如求函数例如求函数y=(3x-2)2的导数的导数,那么我们可以把平方式那么我们可以把平方式 展开展开,利用导数的四则运算法则求导利用导数的四则运算法则求导.然后能否用其它然后能否用其它 的办法求导呢的办法求导呢?又如我们知道函数又如我们知道函数y=1/x2的导数是的导数是 =-2/x3,那么函数那么函数 y=1/(3x-2)2的导数又是什么呢的导数又是什么呢?y 为了解决上面的问题为了解决
2、上面的问题,我们需要学习新的导数的运算我们需要学习新的导数的运算法则法则,这就是这就是复合函数的导数复合函数的导数.二、新课二、新课复合函数的导数：复合函数的导数：1.复合函数的概念复合函数的概念:对于函数对于函数y=f (x),令令u=(x),若若y=f(u)是中间变量是中间变量u的函数的函数,u=(x)是自变量是自变量x的函数的函数,则称则称y=f (x)是自变量是自变量x的复合函数的复合函数.2.复合函数的导数复合函数的导数:设函数设函数 在点在点x处有导数处有导数 ,函数函数y=f(u)在在点点x的对应点的对应点u处有导数处有导数 ,则复合函数则复合函数在点在点x处也有导数处也有导数,
3、且且 或记或记)(xu )(xux )(ufyu )(xfy ;xuxuyy ).()()(xufxfx 如如:求函数求函数y=(3x-2)2的导数的导数,我们就可以有我们就可以有,令令y=u2,u=3x-2,则则 从而从而 .结果与我结果与我们利用导数的四则运算法则求得的结果完全一致们利用导数的四则运算法则求得的结果完全一致.,3,2 xuuuy1218 xuyyxux 在书写时不要把在书写时不要把 写成写成 ,两者是不完两者是不完全一样的全一样的,前者表示对自变量前者表示对自变量x的求导的求导,而后者是对中间而后者是对中间变量变量 的求导的求导.)()(xfxfx )(x 3.复合函数的求
4、导法则复合函数的求导法则:复合函数对自变量的导数复合函数对自变量的导数,等于已知函数对中间等于已知函数对中间变量的导数变量的导数,乘以中间变量对自变量的导数乘以中间变量对自变量的导数.法则可以推广到两个以上的中间变量法则可以推广到两个以上的中间变量.求复合函数的导数求复合函数的导数,关键在于分清函数的复合关关键在于分清函数的复合关系系,合理选定中间变量合理选定中间变量,明确求导过程中每次是哪个变明确求导过程中每次是哪个变量对哪个变量求导量对哪个变量求导,一般地一般地,如果所设中间变量可直接如果所设中间变量可直接求导求导,就不必再选中间变量就不必再选中间变量.复合函数的求导法则与导数的四则运算法
5、则要有复合函数的求导法则与导数的四则运算法则要有机的结合和综合的运用机的结合和综合的运用.要通过求一些初等函数的导要通过求一些初等函数的导数数,逐步掌握复合函数的求导法则逐步掌握复合函数的求导法则.三、例题选讲：三、例题选讲：例例1:求下列函数的导数求下列函数的导数:5)12()1(xy解解:设设y=u5,u=2x+1,则则:.)12(102)12(525)12()(4445 xxuxuuyyxuxux4)31(1)2(xy 解解:设设y=u-4,u=1-3x,则则:.)31(1212)3(4)31()(5554xuuxuuyyxuxux 42)sin1()3(xy 解解:设设y=u-4,u=
6、1+v2,v=sinx,则则:.2sin)sin1(4cossin2)sin1(4cos24)(sin)1()(3232324xxxxxxvuxvuvuyyxvuxvux 说明说明:在对法则的运用熟练后在对法则的运用熟练后,就不必再写中间步骤就不必再写中间步骤.例例2:求下列函数的导数求下列函数的导数:(1)y=(2x3-x+1/x)4;解解:.)116()12(4)12()12(42233333 xxxxxxxxxxxy(3)y=tan3x;解解:.secsin3cos1)cossin(3cos)sin(sincoscos)cossin(3)cossin(tan3)(tan)(tan3422
7、22322xxxxxxxxxxxxxxxxxy (2)51xxy 解解:.)1(51)1(1)1(51)1()1(51565425454xxxxxxxxxy(4)221)32(xxy ;)1)(32(1)32(212222xxxxy 解解:.161)32(142)1(21)32()1(4232222122212xxxxxxxxxxxxxy (5):y=sin2(2x+/3)法一法一:.)324sin(22)32cos()32sin(2 xxxy法二法二:,)324cos(121 xy.)324sin(2 4)324sin(021 xxy练习练习1:求下列函数的导数求下列函数的导数:bxaxyx
8、xyxxxyxycbxaxycossin)5()7643()4()3(211)2()1(232232 答案答案:2223221)21(2)2()(3)2()1(xxxycbxaxcbxaxbaxy 4227421925)76()43(135)4()925()(21)3(xxxxxxy.)2sin()2(41)2sin()2(41sin21)5(xbabaxbababxb 例例3:如果圆的半径以如果圆的半径以2cm/s的等速度增加的等速度增加,求圆半径求圆半径R=10cm时时,圆面积增加的速度圆面积增加的速度.解解:由已知知由已知知:圆半径圆半径R=R(t),且且 =2cm/s.tR 又圆面积又
9、圆面积S=R2,所以所以=40(cm)2/s.2102|2|1010RtRtRRS故圆面积增加的速度为故圆面积增加的速度为40(cm)2/s.例例4:在曲线在曲线 上求一点上求一点,使通过该点的切线平行使通过该点的切线平行于于 x轴轴,并求此切线的方程并求此切线的方程.211xy 解解:设所求点为设所求点为P(x0,y0).则由导数的几何意义知则由导数的几何意义知:切线斜率切线斜率.0,0)1(2|)11()(02200200 xxxxxfkxx把把x0=0代入曲线方程得代入曲线方程得:y0=1.所以点所以点P的坐标为的坐标为(0,1),切线方程为切线方程为y-1=0.例例5:求证双曲线求证双
10、曲线C1:x2-y2=5与椭圆与椭圆C2:4x2+9y2=72在交在交 点处的切线互相垂直点处的切线互相垂直.证证:由于曲线的图形关于坐标轴对称由于曲线的图形关于坐标轴对称,故只需证明其中一故只需证明其中一 个交点处的切线互相垂直即可个交点处的切线互相垂直即可.联立两曲线方程解得第一象限的交点为联立两曲线方程解得第一象限的交点为P(3,2),不妨不妨证明过证明过P点的两条切线互相垂直点的两条切线互相垂直.由于点由于点P在第一象限在第一象限,故由故由x2-y2=5得得,5,522 xxyxy;23|31 xyk同理由同理由4x2+9y2=72得得;94894,94822xxyxy .32|32
11、xyk因为因为k1k2=-1,所以两条切线互相垂直所以两条切线互相垂直.从而命题成立从而命题成立.例例6:设设f(x)可导可导,求下列函数的导数求下列函数的导数:(1)f(x2);(2)f();(3)f(sin2x)+f(cos2x)21x 解解:);(2)()()1(222xf xxxfy );1(1122)1()2(2222xfxxxxxfy ).(cos)(sin2sin)sin(cos2)(coscossin2)(sin)(cos(cos)(sin(sin)(cos)(sin)3(2222222222xfxfxxxxfxxxfxxfxxfxfxfy 说明说明:对于抽象函数的求导对于抽象
12、函数的求导,一方面要从其形式是把握其一方面要从其形式是把握其 结构特征结构特征,另一方面要充分运用复合关系的求导法另一方面要充分运用复合关系的求导法 则则.我们曾经利用导数的定义证明过这样的一个结论我们曾经利用导数的定义证明过这样的一个结论:“可导的偶函数的导函数为奇函数可导的偶函数的导函数为奇函数;可导的奇函数的导函可导的奇函数的导函数为偶函数数为偶函数”.现在我们利用复合函数的现在我们利用复合函数的导数重新加以导数重新加以证明证明:证证:当当f(x)为为可导的偶函数可导的偶函数时时,则则f(-x)=f(x).两边同时对两边同时对x 求导得求导得:,故故 为为 奇函数奇函数.)()()()(
13、xfxfxfxxf )(xf 同理可证另一个命题同理可证另一个命题.我们还可以证明类似的一个结论我们还可以证明类似的一个结论:可导的周期函数可导的周期函数的导函数也是周期函数的导函数也是周期函数.证证:设设f(x)为为可导的周期函数可导的周期函数,T为其一个为其一个周期周期,则对定义则对定义 域内的每一个域内的每一个x,都有都有f(x+T)=f(x).两边同时对两边同时对x求导得求导得:即即 也是以也是以T为为周期的周期函数周期的周期函数.),()(xfTxTxf )(Txf )().(xfxf 例例7:求函数求函数 的导数的导数.11311)(2xxxxxf说明说明:这是分段函数的求导问题这
14、是分段函数的求导问题,先根据各段的函数表达先根据各段的函数表达 式式,求出在各可导求出在各可导(开开)区间的函数的导数区间的函数的导数,然后再用然后再用 定义来讨论分段点的可导性定义来讨论分段点的可导性.解解:当当x1时时,.1312)(xxxxf又又 ,故故f(x)在在x=1处连续处连续.2)1()(lim)(lim11 fxfxfxx而而;2)1(lim121lim1)1()(lim1211 xxxxfxfxxx;33lim12)1(3lim1)1()(lim111 xxxxxxfxf,11)(lim1)1()(lim11 xxfxfxfxx从而从而f(x)在在x=1处不可导处不可导.13
15、12)(xxxxf四、小结：四、小结：利用复合函数的求导法则来求导数时利用复合函数的求导法则来求导数时,选择中间变选择中间变量是复合函数求导的关键量是复合函数求导的关键.必须正确分析复合函数是由必须正确分析复合函数是由哪些基本函数经过怎样的顺序复合而成的哪些基本函数经过怎样的顺序复合而成的,分清其间的分清其间的复合关系复合关系.要善于把一部分量、式子暂时当作一个整体要善于把一部分量、式子暂时当作一个整体,这个暂时的整体这个暂时的整体,就是中间变量就是中间变量.求导时需要记住中间变求导时需要记住中间变量量,注意逐层求导注意逐层求导,不遗漏不遗漏,而其中特别要注意中间变量而其中特别要注意中间变量的
16、系数的系数,求导后求导后,要把中间变量转换成自变量的函数要把中间变量转换成自变量的函数.在上面的例子中涉及到了二次曲线在某点的切线在上面的例子中涉及到了二次曲线在某点的切线问题问题,但在上面的解法中回避了点在第二、三、四象限但在上面的解法中回避了点在第二、三、四象限的情况的情况.可能有同学会提出对于二次曲线在任意点的切可能有同学会提出对于二次曲线在任意点的切线怎样求的问题线怎样求的问题,由于它涉及到隐函数的求导问题由于它涉及到隐函数的求导问题.我们我们不便去过多的去研究不便去过多的去研究.下面举一个例子使同学们了解一下求一般曲线在任下面举一个例子使同学们了解一下求一般曲线在任意点的切线的方法意
17、点的切线的方法.(说明说明:这个内容不属于考查范围这个内容不属于考查范围.)例子例子:求椭圆求椭圆 在点在点 处的切线方程处的切线方程.191622 yx)323,2(解解:对椭圆方程的两边分别求导对椭圆方程的两边分别求导(在此把在此把y看成是关于看成是关于x 的函数的函数)得得:.169,02912161yxyyyx .43|2323 xyyk于是所求切线方程为于是所求切线方程为:.03843),2(43233 yxxy即即备用备用利用上述方法可得圆锥曲线的切线方程如下利用上述方法可得圆锥曲线的切线方程如下:(1)过圆过圆(x-a)2+(y-b)2=r2上一点上一点P0(x0,y0)的切线方
18、程是的切线方程是:(x0-a)(x-a)+(y0-b)(y-b)=r2.(2)过椭圆过椭圆 上一点上一点P0(x0,y0)的切线方程是的切线方程是:12222 byax.12020 byyaxx(2)过椭圆过椭圆 上一点上一点P0(x0,y0)的切线方程是的切线方程是:(4)过抛物线过抛物线y2=2px上一点上一点P0(x0,y0)的切线方程是的切线方程是:y0y =p(x+x0).12020 byyaxx(3)过双曲线过双曲线 上一点上一点P0(x0,y0)的切线方程是的切线方程是:12222 byax证证:设设x有增量有增量x,则对应的则对应的u,y分别有增量分别有增量u,y.因为因为 在
19、点在点x处可导处可导,所以所以 在点在点x处连续处连续.因此当因此当x 0时时,u 0.)(xu )(xu 当当u0时时,由由 ,且且 得得:xuuyxy uyuyux 00limlim.,limlimlimlimlim00000 xuxxuxxxuyyxuuyxuuyxy 即即当当u=0时时,公式也成立公式也成立.上面的证明其实不是一个很严格的证明上面的证明其实不是一个很严格的证明,而且中间而且中间还会有不少的疑问还会有不少的疑问,譬如譬如,u=0时公式也成立时公式也成立,怎样去理怎样去理解解;x 0时与时与u 0时的极限相等问题等等时的极限相等问题等等.因此同学因此同学们只要了解公式证明中
20、的基本思想和方法即可们只要了解公式证明中的基本思想和方法即可,不必过不必过多的去深究证明的过程多的去深究证明的过程.因为事实上因为事实上,还有更严格的证明还有更严格的证明.初一英语第二学期同步串讲初一英语第二学期同步串讲 第七讲第七讲 Do you have an eraser?Do you have a soccer ball?soccer basketballbadmintonbaseballball gamestennis(racket)volleyballgolfPing pong(ball)table tennis Do you have an American football/r
21、ugby?Yes,I do./No,I dont.Golf clubDo you have a golf ball?Yes,I do./No,I dont.Group workNames Do you have a?Balls you haveLucyYes,I do.basketballWhat balls do you have?I have.1.你有一个排球吗？你有一个排球吗？是的，我有。是的，我有。2.你有表兄妹吗你有表兄妹吗?不不,我没有。我没有。3.她有网球拍吗？她有网球拍吗？是的，她有。是的，她有。7.咱们去踢球吧。咱们去踢球吧。Do you have a volleyball?
22、Yes,I do.Do you have any cousins?No,I dont.Does she have a tennis racket?Yes,she does.Lets play soccer.4.他有英英字典吗？他有英英字典吗？不，他没有。他有一不，他没有。他有一本英汉字典。本英汉字典。5.他们有笔记本吗？他们有笔记本吗？是的，他们有。是的，他们有。6.让我看一看。让我看一看。Does he have an English English dictionary?No,he doesnt.He has an English Chinese dictionary?Do they ha
23、ve any note books?Yes,they do.Lets have a look.Does he have a tennis racket?Yes,he does./No,he doesnt.Does she have a soccer ball?Yes,she does./No,she doesnt.Does he have a basketball?Yes,he does.Do they have a puck?Yes,they do./No,they dont.ice hockeyA:I want to play baseball.Do you have a baseball
24、 bat?B:No,I dont.A:Does your brother have one?B:Yes,he does.Lets go and find him.Lets play volleyball.That sounds good.Lets play barbell.BarbellThat sounds boring.Lets play bowling.That sounds interesting.Lets play baseball.That sounds difficult.Lets go skateboarding.That sounds fun.(exciting)Lets p
25、lay Ping Pong.That sounds relaxing.let sb.do sth.Lets have a break.Lets go and play Ping-pong.Our School Sports Center needs new members.Fill in the enrollment form if you want to join the clubs.First name _ Last name _Age _ Sex _Class _ Club _Reason _Whats your first name,please?How old are you?Wha
26、t class are you in?What club do you want to join?Why?(Because I like)写出反义词写出反义词 dislike_ interesting _ good_ well_ easy_ lost _ day _ different _ big _ first _ on _ in front of _ this _ these _ bring _ Fill in the blanks1.A:Excuse me,_ you _ a baseball?B:No,I _.A:_ Jim _ one?B:Yes,he _.Lets go and f
27、ind him.2.A:Hi,Sally.B:Hi,Jane.A:Sally,_ _ my friend,Lucy.B:Hi,Lucy._ to meet you.C:Nice to _ you,_.do havedontDoes havedoesthis isNice meet too A:Lets play volleyball.B:That _ interesting,but I _ have one.A:Well,_ you have a tennis bat?B:Yes,I _.Lets go and play tennis.A:Thats a good _.soundsdontdo
28、doidea1.Does B have an eraser?2.Is the shop open?3.What day is today?4.What can they do?No,she doesnt.No,it isnt.It is closed.Today is Friday.They go to the supermarket.Part 11.What would B like?2.How much money does B have?3.Where is her purse?4.What does she do?She only has two yuan.Her purse is a
29、t home.She borrows four yuan from C.Part 2Shed like a red pencil and a pencil-sharpener.1.Does Jim have a dictionary?2.Does Ann have a dictionary?3.Who has an English-Chinese dictionary?4.When can Li Lei give it back?Part 3No,he doesnt.Lucy has one.He can give it back next Monday.No,she doesnt.close
30、 *closed 1.The shop closes at 8 pm.2.Please close the door.3.The door is closed.4.The shop is closed.5.The shop is open.borrow from 1.I borrow a book from him.2.May I borrow a pen from you?3.Dont borrow money from her.May I 1.May I come in?2.May I borrow a pen from you?3.May I sit down?give back (to
31、 sb.)1.Please give the money back tomorrow.2.I can give the book back to him next week.3.Can you give my dictionary back this afternoon?There is something wrong with1.There is something wrong with my TV/his pen/your computer.2.There is something wrong with him/her/you/the boy.3.Whats wrong with you?
32、4.Whats wrong with your TV?close *closed borrow from May I give back (to sb.)There is something wrong withYou are going to have a P.E class,think about some activities.Talk about them.Time _ Place _ Sport _ Reason _Writing Write a passage about yourself and your friend.What sports do you play?What s
33、ports things do you have?What other things do you do?What about your friend?(parent)Key points:Key points:Food Do you like bananas?Does he/she like bananas?Do you like bananas?Do you like bananas?Do you like oranges?Yes,I do.No,I dont.Does he like apples?Yes,he does.No,he doesnt.Does she like peache
34、s?Yes,she does.No,she doesnt.Do they like grapes?Yes,they do.No,they dont.I like hamburgers for breakfast.But I dont like eggs.Do your parents Do your parents like pizza?like pizza?Yes,they do.Yes,they do.No,they dont.No,they dont.Does your father Does your father like beer?like beer?Yes,he does.Yes
35、,he does.No,he doesnt.No,he doesnt.Choose the right words.1.There are five _(tomatoes,tomato)on the table.2.The _(child,children)are in the room.3.She _(doesnt like,dont like)chicken for lunch.4.He _(likes,like)hamburgers.5.Do they_(likes,like)apples?6.Mike doesnt like_(bananas,banana).7.My mother d
36、oesnt have _(a lunch,lunch)at home.8.Does she _(like,likes)French fries?用所给动词的适当形式填空。用所给动词的适当形式填空。1.She _(not like)salad.2._you _(like)oranges?3.He _(eat)lots of healthy food every day.4.There _(be)some chicken on the plate.5.His mother_(like)ice cream.6._your father _(like)eggs for breakfast?7.They
37、_(not like)playing tennis.8.There_(be)some potatoes in the bag.9.My mother_(have)supper at home every day.10.Mr.Green_(not have)lunch at school every day.A:Do you want to go to a movie?B:Yes,I do.That sounds good.A:What kind of movies do you like?B:I like action movies because they are exciting.And
38、I think cartoons are interesting.A:How about thrillers?B:No.Thrillers are scary.How about you?A:I dont like thrillers,either.But I like documentaries.I think theyre instructive.Does he want to go to a movie?Yes,he does./No,he doesnt.What kind of movies does he want to see?He wants to see a comedy.What do you like to do on holidays?Why do you like?
