復(fù)合函數(shù)求導(dǎo)教學(xué)課件

復(fù)合函數(shù)的導(dǎo)數(shù)復(fù)合函數(shù)的導(dǎo)數(shù)一、復(fù)習(xí)與引入：1.函數(shù)的導(dǎo)數(shù)的定義與幾何意義.2.常見函數(shù)的導(dǎo)數(shù)公式.3.導(dǎo)數(shù)的四則運(yùn)算法則.4.例如求函數(shù)y=(3x-2)2的導(dǎo)數(shù),那么我們可以把平方式展開,利用導(dǎo)數(shù)的四則運(yùn)算法則求導(dǎo).然后能否用其它的辦法求導(dǎo)呢?又如我們知道函數(shù)y=1/x2的導(dǎo)數(shù)是=-2/x3,那么函數(shù)

y=1/(3x-2)2的導(dǎo)數(shù)又是什么呢?為了解決上面的問題,我們需要學(xué)習(xí)新的導(dǎo)數(shù)的運(yùn)算法則,這就是復(fù)合函數(shù)的導(dǎo)數(shù).一、復(fù)習(xí)與引入：1.函數(shù)的導(dǎo)數(shù)的定義與幾何意義.2.常見函二、新課——復(fù)合函數(shù)的導(dǎo)數(shù)：1.復(fù)合函數(shù)的概念:對于函數(shù)y=f[(x)],令u=(x),若y=f(u)是中間變量u的函數(shù),u=(x)是自變量x的函數(shù),則稱y=f[(x)]是自變量x的復(fù)合函數(shù).2.復(fù)合函數(shù)的導(dǎo)數(shù):設(shè)函數(shù)在點(diǎn)x處有導(dǎo)數(shù),函數(shù)y=f(u)在點(diǎn)x的對應(yīng)點(diǎn)u處有導(dǎo)數(shù),則復(fù)合函數(shù)在點(diǎn)x處也有導(dǎo)數(shù),且或記如:求函數(shù)y=(3x-2)2的導(dǎo)數(shù),我們就可以有,令y=u2,u=3x-2,則從而.結(jié)果與我們利用導(dǎo)數(shù)的四則運(yùn)算法則求得的結(jié)果完全一致.二、新課——復(fù)合函數(shù)的導(dǎo)數(shù)：1.復(fù)合函數(shù)的概念:對于函數(shù)y=

在書寫時不要把寫成,兩者是不完全一樣的,前者表示對自變量x的求導(dǎo),而后者是對中間變量的求導(dǎo).3.復(fù)合函數(shù)的求導(dǎo)法則:

復(fù)合函數(shù)對自變量的導(dǎo)數(shù),等于已知函數(shù)對中間變量的導(dǎo)數(shù),乘以中間變量對自變量的導(dǎo)數(shù).法則可以推廣到兩個以上的中間變量.

求復(fù)合函數(shù)的導(dǎo)數(shù),關(guān)鍵在于分清函數(shù)的復(fù)合關(guān)系,合理選定中間變量,明確求導(dǎo)過程中每次是哪個變量對哪個變量求導(dǎo),一般地,如果所設(shè)中間變量可直接求導(dǎo),就不必再選中間變量.

復(fù)合函數(shù)的求導(dǎo)法則與導(dǎo)數(shù)的四則運(yùn)算法則要有機(jī)的結(jié)合和綜合的運(yùn)用.要通過求一些初等函數(shù)的導(dǎo)數(shù),逐步掌握復(fù)合函數(shù)的求導(dǎo)法則.在書寫時不要把寫成三、例題選講：例1:求下列函數(shù)的導(dǎo)數(shù):解:設(shè)y=u5,u=2x+1,則:解:設(shè)y=u-4,u=1-3x,則:解:設(shè)y=u-4,u=1+v2,v=sinx,則:說明:在對法則的運(yùn)用熟練后,就不必再寫中間步驟.三、例題選講：例1:求下列函數(shù)的導(dǎo)數(shù):解:設(shè)y=u5,u=2例2:求下列函數(shù)的導(dǎo)數(shù):(1)y=(2x3-x+1/x)4;解:(3)y=tan3x;解:(2)解:(4)解:例2:求下列函數(shù)的導(dǎo)數(shù):(1)y=(2x3-x+1/x)4;(5):y=sin2(2x+π/3)法一:法二:練習(xí)1:求下列函數(shù)的導(dǎo)數(shù):答案:(5):y=sin2(2x+π/3)法一:法二:練習(xí)1:求下例3:如果圓的半徑以2cm/s的等速度增加,求圓半徑R=10cm時,圓面積增加的速度.解:由已知知:圓半徑R=R(t),且=2cm/s.又圓面積S=πR2,所以=40π(cm)2/s.故圓面積增加的速度為40π(cm)2/s.例4:在曲線上求一點(diǎn),使通過該點(diǎn)的切線平行于

x軸,并求此切線的方程.解:設(shè)所求點(diǎn)為P(x0,y0).則由導(dǎo)數(shù)的幾何意義知:切線斜率把x0=0代入曲線方程得:y0=1.所以點(diǎn)P的坐標(biāo)為(0,1),切線方程為y-1=0.例3:如果圓的半徑以2cm/s的等速度增加,求圓半徑R=解:例5:求證雙曲線C1:x2-y2=5與橢圓C2:4x2+9y2=72在交點(diǎn)處的切線互相垂直.證:由于曲線的圖形關(guān)于坐標(biāo)軸對稱,故只需證明其中一個交點(diǎn)處的切線互相垂直即可.聯(lián)立兩曲線方程解得第一象限的交點(diǎn)為P(3,2),不妨證明過P點(diǎn)的兩條切線互相垂直.由于點(diǎn)P在第一象限,故由x2-y2=5得同理由4x2+9y2=72得因?yàn)閗1k2=-1,所以兩條切線互相垂直.從而命題成立.例5:求證雙曲線C1:x2-y2=5與橢圓C2:4x2+9y例6:設(shè)f(x)可導(dǎo),求下列函數(shù)的導(dǎo)數(shù):(1)f(x2);(2)f();(3)f(sin2x)+f(cos2x)解:說明:對于抽象函數(shù)的求導(dǎo),一方面要從其形式是把握其結(jié)構(gòu)特征,另一方面要充分運(yùn)用復(fù)合關(guān)系的求導(dǎo)法則.例6:設(shè)f(x)可導(dǎo),求下列函數(shù)的導(dǎo)數(shù):解:說明:對于抽象

我們曾經(jīng)利用導(dǎo)數(shù)的定義證明過這樣的一個結(jié)論:“可導(dǎo)的偶函數(shù)的導(dǎo)函數(shù)為奇函數(shù);可導(dǎo)的奇函數(shù)的導(dǎo)函數(shù)為偶函數(shù)”.現(xiàn)在我們利用復(fù)合函數(shù)的導(dǎo)數(shù)重新加以證明:證:當(dāng)f(x)為可導(dǎo)的偶函數(shù)時,則f(-x)=f(x).兩邊同時對x

求導(dǎo)得:,故為奇函數(shù).同理可證另一個命題.

我們還可以證明類似的一個結(jié)論:可導(dǎo)的周期函數(shù)的導(dǎo)函數(shù)也是周期函數(shù).證:設(shè)f(x)為可導(dǎo)的周期函數(shù),T為其一個周期,則對定義域內(nèi)的每一個x,都有f(x+T)=f(x).

兩邊同時對x求導(dǎo)得:即也是以T為周期的周期函數(shù).我們曾經(jīng)利用導(dǎo)數(shù)的定義證明過這樣的一個結(jié)論:證例7:求函數(shù)的導(dǎo)數(shù).說明:這是分段函數(shù)的求導(dǎo)問題,先根據(jù)各段的函數(shù)表達(dá)式,求出在各可導(dǎo)(開)區(qū)間的函數(shù)的導(dǎo)數(shù),然后再用定義來討論分段點(diǎn)的可導(dǎo)性.解:當(dāng)x≠1時,.又,故f(x)在x=1處連續(xù).而從而f(x)在x=1處不可導(dǎo).例7:求函數(shù)四、小結(jié)：

利用復(fù)合函數(shù)的求導(dǎo)法則來求導(dǎo)數(shù)時,選擇中間變量是復(fù)合函數(shù)求導(dǎo)的關(guān)鍵.必須正確分析復(fù)合函數(shù)是由哪些基本函數(shù)經(jīng)過怎樣的順序復(fù)合而成的,分清其間的復(fù)合關(guān)系.要善于把一部分量、式子暫時當(dāng)作一個整體,這個暫時的整體,就是中間變量.求導(dǎo)時需要記住中間變量,注意逐層求導(dǎo),不遺漏,而其中特別要注意中間變量的系數(shù),求導(dǎo)后,要把中間變量轉(zhuǎn)換成自變量的函數(shù).四、小結(jié)：利用復(fù)合函數(shù)的求導(dǎo)法則來求導(dǎo)數(shù)時,選

在上面的例子中涉及到了二次曲線在某點(diǎn)的切線問題,但在上面的解法中回避了點(diǎn)在第二、三、四象限的情況.可能有同學(xué)會提出對于二次曲線在任意點(diǎn)的切線怎樣求的問題,由于它涉及到隱函數(shù)的求導(dǎo)問題.我們不便去過多的去研究.

下面舉一個例子使同學(xué)們了解一下求一般曲線在任意點(diǎn)的切線的方法.(說明:這個內(nèi)容不屬于考查范圍.)例子:求橢圓在點(diǎn)處的切線方程.解:對橢圓方程的兩邊分別求導(dǎo)(在此把y看成是關(guān)于x

的函數(shù))得:于是所求切線方程為:備用在上面的例子中涉及到了二次曲線在某點(diǎn)的切線利用上述方法可得圓錐曲線的切線方程如下:(1)過圓(x-a)2+(y-b)2=r2上一點(diǎn)P0(x0,y0)的切線方程是:(x0-a)(x-a)+(y0-b)(y-b)=r2.(2)過橢圓上一點(diǎn)P0(x0,y0)的切線方程是:(2)過橢圓上一點(diǎn)P0(x0,y0)的切線方程是:(4)過拋物線y2=2px上一點(diǎn)P0(x0,y0)的切線方程是:y0y=p(x+x0).(3)過雙曲線上一點(diǎn)P0(x0,y0)的切線方程是:利用上述方法可得圓錐曲線的切線方程如下:(1)過圓(x-a)證:設(shè)x有增量Δx,則對應(yīng)的u,y分別有增量Δu,Δy.因?yàn)樵邳c(diǎn)x處可導(dǎo),所以在點(diǎn)x處連續(xù).因此當(dāng)Δx→0時,Δu→0.當(dāng)Δu≠0時,由,且得:當(dāng)Δu=0時,公式也成立.

上面的證明其實(shí)不是一個很嚴(yán)格的證明,而且中間還會有不少的疑問,譬如,Δu=0時公式也成立,怎樣去理解;Δx→0時與Δu→0時的極限相等問題等等.因此同學(xué)們只要了解公式證明中的基本思想和方法即可,不必過多的去深究證明的過程.因?yàn)槭聦?shí)上,還有更嚴(yán)格的證明.證:設(shè)x有增量Δx,則對應(yīng)的u,y分別有增量Δu,Δy.因初一英語第二學(xué)期同步串講第七講

Doyouhaveaneraser?初一英語第二學(xué)期同步串講Doyouhaveasoccerball?復(fù)合函數(shù)求導(dǎo)教學(xué)課件soccerbasketballbadmintonbaseballballgamessoccerbasketballbadmintonbasetennis(racket)volleyballgolfPingpong(ball)tabletennistennis(racket)volleyballgolfPDoyouhaveanAmericanfootball/rugby?Yes,Ido./No,Idon’t.DoyouhaveanAmericanfootbaGolfclubDoyouhaveagolfball?Yes,Ido./No,Idon’t.GolfclubDoyouhaveagolfbaGroupworkNamesDoyouhavea…?BallsyouhaveLucyYes,Ido.basketballWhatballsdoyouhave?Ihave….GroupworkNamesDoyouhavea…1.你有一個排球嗎？是的，我有。2.你有表兄妹嗎?不,我沒有。3.她有網(wǎng)球拍嗎？是的，她有。7.咱們?nèi)ヌ咔虬?。Doyouhaveavolleyball?Yes,Ido.Doyouhaveanycousins?No,Idon’t.Doesshehaveatennisracket?Yes,shedoes.Let’splaysoccer.1.你有一個排球嗎？是的，我有。Doyou4.他有英英字典嗎？不，他沒有。他有一本英漢字典。5.他們有筆記本嗎？是的，他們有。6.讓我看一看。DoeshehaveanEnglishEnglishdictionary?No,hedoesn’t.HehasanEnglishChinesedictionary?Dotheyhaveanynotebooks?Yes,theydo.Let’shavealook.4.他有英英字典嗎？不，他沒有。他有一本英漢字典。DoeDoeshehaveatennisracket?Yes,hedoes./No,hedoesn't.Doeshehaveatennisracket?YDoesshehaveasoccerball?Yes,shedoes./No,shedoesn't.Doesshehaveasoccerball?YeDoeshehaveabasketball?Yes,hedoes.Doeshehaveabasketball?Yes,Dotheyhaveapuck?Yes,theydo./No,theydon’t.icehockeyDotheyhaveapuck?icehockeyA:Iwanttoplaybaseball.Doyouhaveabaseballbat?B:No,Idon’t.A:Doesyourbrotherhaveone?B:Yes,hedoes.Let’sgoandfindhim.A:Iwanttoplaybaseball.DoLet’splayvolleyball.Thatsoundsgood.Let’splayvolleyball.Let’splaybarbell.BarbellThatsoundsboring.Let’splaybarbell.BarbellThatLet’splaybowling.Thatsoundsinteresting.Let’splaybowling.ThatsoundsLet’splaybaseball.Thatsoundsdifficult.Let’splaybaseball.Let’sgoskateboarding.Thatsoundsfun.(exciting)Let’sgoskateboarding.Let’splayPingPong.Thatsoundsrelaxing.letsb.dosth.Let’shaveabreak.Let’sgoandplayPing-pong.Let’splayPingPong.letsb.dOurSchoolSportsCenterneedsnewmembers.Fillintheenrollmentformifyouwanttojointheclubs.Firstname__________Lastname__________Age________________Sex________________Class______________Club_______________Reason__________________________________What’syourfirstname,please?Howoldareyou?Whatclassareyouin?Whatclubdoyouwanttojoin?Why?(Because…Ilike…)OurSchoolSportsCenterneed寫出反義詞dislike______interesting______good_______well________easy________lost_________day________different_______big________first_________on________infrontof_________this_______these_____________bring______寫出反義詞Fillintheblanks1.A:Excuseme,_____you____abaseball?B:No,I_____.A:______Jim_____one?B:Yes,he______.Let’sgoandfindhim.2.A:Hi,Sally.B:Hi,Jane.A:Sally,________myfriend,Lucy.B:Hi,Lucy._____tomeetyou.C:Niceto_____you,_____.dohavedon’tDoeshavedoesthisisNicemeettooFillintheblanksdoA:Let’splayvolleyball.B:That______interesting,butI_____haveone.A:Well,_____youhaveatennisbat?B:Yes,I_____.Let’sgoandplaytennis.A:That’sagood______.soundsdon’tdodoideaA:Let’splayvolleyball.soundDoesBhaveaneraser?Istheshopopen?Whatdayistoday?Whatcantheydo?No,shedoesn’t.No,itisn’t.Itisclosed.TodayisFriday.Theygotothesupermarket.Part1DoesBhaveaneraser?No,sheWhatwouldBlike?HowmuchmoneydoesBhave?Whereisherpurse?Whatdoesshedo?

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Thereissomethingwrongwith…close***closedborrYouaregoingtohaveaP.Eclass,thinkaboutsomeactivities.Talkaboutthem.Time_________Place________Sport________Reason________YouaregoingtohaveaP.EclWritingWriteapassageaboutyourselfandyourfriend.Whatsportsdoyouplay?Whatsportsthingsdoyouhave?Whatotherthingsdoyoudo?Whataboutyourfriend?(parent)WritingWriteapassageaboutyKeypoints:

FoodDoyoulikebananas?Doeshe/shelikebananas?Doyoulikebananas?Keypoints:FoodDoyoulikebDoyoulikeoranges?Yes,Ido.No,Idon’t.Doyoulikeoranges?Doeshelikeapples?Yes,hedoes.No,hedoesn't.Doeshelikeapples?Doesshelikepeaches?Yes,shedoes.No,shedoesn't.Doesshelikepeaches?Dotheylikegrapes?Yes,theydo.No,theydon’t.Dotheylikegrapes?Ilikehamburgersforbreakfast.ButIdon’tlikeeggs.IlikehamburgersButIdon’tlDoyourparentslikepizza?Yes,theydo.No,theydon’t.

Doyourparentslikepizza?Doesyourfatherlikebeer?Yes,hedoes.No,hedoesn't.Doesyourfatherlikebeer?Choosetherightwords.Therearefive_______(tomatoes,tomato)onthetable.The______(child,children)areintheroom.She_______(doesn’tlike,don’tlike)chickenforlunch.He______(likes,like)hamburgers.Choosetherightwords.5.Dothey_____(likes,like)apples?6.Mikedoesn’tlike______(bananas,banana).7.Mymotherdoesn’thave______(alunch,lunch)athome.8.Doesshe______(like,likes)Frenchfries?5.Dothey_____(likes,like)a用所給動詞的適當(dāng)形式填空。She______(notlike)salad._____you_____(like)oranges?He_____(eat)lotsofhealthyfoodeveryday.There_____(be)somechickenontheplate.Hismother_____(like)icecream.用所給動詞的適當(dāng)形式填空。6.______yourfather_____(like)eggsforbreakfast?7.They_____(notlike)playingtennis.8.There_____(be)somepotatoesinthebag.9.Mymother_____(have)supperathomeeveryday.10.Mr.Green_____(nothave)lunchatschooleveryday.6.______yourfather_____(likA:Doyouwanttogotoamovie?B:Yes,Ido.Thatsoundsgood.A:Whatkindofmoviesdoyoulike?B:Ilikeactionmoviesbecausetheyareexciting.AndIthinkcartoonsareinteresting.A:Howaboutthrillers?B:No.Thrillersarescary.Howaboutyou?A:Idon’tlikethrillers,either.ButIlikedocumentaries.Ithinkthey’reinstructive.A:DoyouwanttogotoamoviDoeshewanttogotoamovie?Yes,hedoes./No,hedoesn’t.Whatkindofmoviesdoeshewanttosee?Hewantstoseeacomedy.Doeshewanttogotoamovie?Whatdoyouliketodoonholidays?Whydoyoulike…?Whatdoyouliketodo復(fù)合函數(shù)的導(dǎo)數(shù)復(fù)合函數(shù)的導(dǎo)數(shù)一、復(fù)習(xí)與引入：1.函數(shù)的導(dǎo)數(shù)的定義與幾何意義.2.常見函數(shù)的導(dǎo)數(shù)公式.3.導(dǎo)數(shù)的四則運(yùn)算法則.4.例如求函數(shù)y=(3x-2)2的導(dǎo)數(shù),那么我們可以把平方式展開,利用導(dǎo)數(shù)的四則運(yùn)算法則求導(dǎo).然后能否用其它的辦法求導(dǎo)呢?又如我們知道函數(shù)y=1/x2的導(dǎo)數(shù)是=-2/x3,那么函數(shù)

y=1/(3x-2)2的導(dǎo)數(shù)又是什么呢?為了解決上面的問題,我們需要學(xué)習(xí)新的導(dǎo)數(shù)的運(yùn)算法則,這就是復(fù)合函數(shù)的導(dǎo)數(shù).一、復(fù)習(xí)與引入：1.函數(shù)的導(dǎo)數(shù)的定義與幾何意義.2.常見函二、新課——復(fù)合函數(shù)的導(dǎo)數(shù)：1.復(fù)合函數(shù)的概念:對于函數(shù)y=f[(x)],令u=(x),若y=f(u)是中間變量u的函數(shù),u=(x)是自變量x的函數(shù),則稱y=f[(x)]是自變量x的復(fù)合函數(shù).2.復(fù)合函數(shù)的導(dǎo)數(shù):設(shè)函數(shù)在點(diǎn)x處有導(dǎo)數(shù),函數(shù)y=f(u)在點(diǎn)x的對應(yīng)點(diǎn)u處有導(dǎo)數(shù),則復(fù)合函數(shù)在點(diǎn)x處也有導(dǎo)數(shù),且或記如:求函數(shù)y=(3x-2)2的導(dǎo)數(shù),我們就可以有,令y=u2,u=3x-2,則從而.結(jié)果與我們利用導(dǎo)數(shù)的四則運(yùn)算法則求得的結(jié)果完全一致.二、新課——復(fù)合函數(shù)的導(dǎo)數(shù)：1.復(fù)合函數(shù)的概念:對于函數(shù)y=

在書寫時不要把寫成,兩者是不完全一樣的,前者表示對自變量x的求導(dǎo),而后者是對中間變量的求導(dǎo).3.復(fù)合函數(shù)的求導(dǎo)法則:

復(fù)合函數(shù)對自變量的導(dǎo)數(shù),等于已知函數(shù)對中間變量的導(dǎo)數(shù),乘以中間變量對自變量的導(dǎo)數(shù).法則可以推廣到兩個以上的中間變量.

求復(fù)合函數(shù)的導(dǎo)數(shù),關(guān)鍵在于分清函數(shù)的復(fù)合關(guān)系,合理選定中間變量,明確求導(dǎo)過程中每次是哪個變量對哪個變量求導(dǎo),一般地,如果所設(shè)中間變量可直接求導(dǎo),就不必再選中間變量.

復(fù)合函數(shù)的求導(dǎo)法則與導(dǎo)數(shù)的四則運(yùn)算法則要有機(jī)的結(jié)合和綜合的運(yùn)用.要通過求一些初等函數(shù)的導(dǎo)數(shù),逐步掌握復(fù)合函數(shù)的求導(dǎo)法則.在書寫時不要把寫成三、例題選講：例1:求下列函數(shù)的導(dǎo)數(shù):解:設(shè)y=u5,u=2x+1,則:解:設(shè)y=u-4,u=1-3x,則:解:設(shè)y=u-4,u=1+v2,v=sinx,則:說明:在對法則的運(yùn)用熟練后,就不必再寫中間步驟.三、例題選講：例1:求下列函數(shù)的導(dǎo)數(shù):解:設(shè)y=u5,u=2例2:求下列函數(shù)的導(dǎo)數(shù):(1)y=(2x3-x+1/x)4;解:(3)y=tan3x;解:(2)解:(4)解:例2:求下列函數(shù)的導(dǎo)數(shù):(1)y=(2x3-x+1/x)4;(5):y=sin2(2x+π/3)法一:法二:練習(xí)1:求下列函數(shù)的導(dǎo)數(shù):答案:(5):y=sin2(2x+π/3)法一:法二:練習(xí)1:求下例3:如果圓的半徑以2cm/s的等速度增加,求圓半徑R=10cm時,圓面積增加的速度.解:由已知知:圓半徑R=R(t),且=2cm/s.又圓面積S=πR2,所以=40π(cm)2/s.故圓面積增加的速度為40π(cm)2/s.例4:在曲線上求一點(diǎn),使通過該點(diǎn)的切線平行于

x軸,并求此切線的方程.解:設(shè)所求點(diǎn)為P(x0,y0).則由導(dǎo)數(shù)的幾何意義知:切線斜率把x0=0代入曲線方程得:y0=1.所以點(diǎn)P的坐標(biāo)為(0,1),切線方程為y-1=0.例3:如果圓的半徑以2cm/s的等速度增加,求圓半徑R=解:例5:求證雙曲線C1:x2-y2=5與橢圓C2:4x2+9y2=72在交點(diǎn)處的切線互相垂直.證:由于曲線的圖形關(guān)于坐標(biāo)軸對稱,故只需證明其中一個交點(diǎn)處的切線互相垂直即可.聯(lián)立兩曲線方程解得第一象限的交點(diǎn)為P(3,2),不妨證明過P點(diǎn)的兩條切線互相垂直.由于點(diǎn)P在第一象限,故由x2-y2=5得同理由4x2+9y2=72得因?yàn)閗1k2=-1,所以兩條切線互相垂直.從而命題成立.例5:求證雙曲線C1:x2-y2=5與橢圓C2:4x2+9y例6:設(shè)f(x)可導(dǎo),求下列函數(shù)的導(dǎo)數(shù):(1)f(x2);(2)f();(3)f(sin2x)+f(cos2x)解:說明:對于抽象函數(shù)的求導(dǎo),一方面要從其形式是把握其結(jié)構(gòu)特征,另一方面要充分運(yùn)用復(fù)合關(guān)系的求導(dǎo)法則.例6:設(shè)f(x)可導(dǎo),求下列函數(shù)的導(dǎo)數(shù):解:說明:對于抽象

我們曾經(jīng)利用導(dǎo)數(shù)的定義證明過這樣的一個結(jié)論:“可導(dǎo)的偶函數(shù)的導(dǎo)函數(shù)為奇函數(shù);可導(dǎo)的奇函數(shù)的導(dǎo)函數(shù)為偶函數(shù)”.現(xiàn)在我們利用復(fù)合函數(shù)的導(dǎo)數(shù)重新加以證明:證:當(dāng)f(x)為可導(dǎo)的偶函數(shù)時,則f(-x)=f(x).兩邊同時對x

求導(dǎo)得:,故為奇函數(shù).同理可證另一個命題.

我們還可以證明類似的一個結(jié)論:可導(dǎo)的周期函數(shù)的導(dǎo)函數(shù)也是周期函數(shù).證:設(shè)f(x)為可導(dǎo)的周期函數(shù),T為其一個周期,則對定義域內(nèi)的每一個x,都有f(x+T)=f(x).

兩邊同時對x求導(dǎo)得:即也是以T為周期的周期函數(shù).我們曾經(jīng)利用導(dǎo)數(shù)的定義證明過這樣的一個結(jié)論:證例7:求函數(shù)的導(dǎo)數(shù).說明:這是分段函數(shù)的求導(dǎo)問題,先根據(jù)各段的函數(shù)表達(dá)式,求出在各可導(dǎo)(開)區(qū)間的函數(shù)的導(dǎo)數(shù),然后再用定義來討論分段點(diǎn)的可導(dǎo)性.解:當(dāng)x≠1時,.又,故f(x)在x=1處連續(xù).而從而f(x)在x=1處不可導(dǎo).例7:求函數(shù)四、小結(jié)：

利用復(fù)合函數(shù)的求導(dǎo)法則來求導(dǎo)數(shù)時,選擇中間變量是復(fù)合函數(shù)求導(dǎo)的關(guān)鍵.必須正確分析復(fù)合函數(shù)是由哪些基本函數(shù)經(jīng)過怎樣的順序復(fù)合而成的,分清其間的復(fù)合關(guān)系.要善于把一部分量、式子暫時當(dāng)作一個整體,這個暫時的整體,就是中間變量.求導(dǎo)時需要記住中間變量,注意逐層求導(dǎo),不遺漏,而其中特別要注意中間變量的系數(shù),求導(dǎo)后,要把中間變量轉(zhuǎn)換成自變量的函數(shù).四、小結(jié)：利用復(fù)合函數(shù)的求導(dǎo)法則來求導(dǎo)數(shù)時,選

在上面的例子中涉及到了二次曲線在某點(diǎn)的切線問題,但在上面的解法中回避了點(diǎn)在第二、三、四象限的情況.可能有同學(xué)會提出對于二次曲線在任意點(diǎn)的切線怎樣求的問題,由于它涉及到隱函數(shù)的求導(dǎo)問題.我們不便去過多的去研究.

下面舉一個例子使同學(xué)們了解一下求一般曲線在任意點(diǎn)的切線的方法.(說明:這個內(nèi)容不屬于考查范圍.)例子:求橢圓在點(diǎn)處的切線方程.解:對橢圓方程的兩邊分別求導(dǎo)(在此把y看成是關(guān)于x

的函數(shù))得:于是所求切線方程為:備用在上面的例子中涉及到了二次曲線在某點(diǎn)的切線利用上述方法可得圓錐曲線的切線方程如下:(1)過圓(x-a)2+(y-b)2=r2上一點(diǎn)P0(x0,y0)的切線方程是:(x0-a)(x-a)+(y0-b)(y-b)=r2.(2)過橢圓上一點(diǎn)P0(x0,y0)的切線方程是:(2)過橢圓上一點(diǎn)P0(x0,y0)的切線方程是:(4)過拋物線y2=2px上一點(diǎn)P0(x0,y0)的切線方程是:y0y=p(x+x0).(3)過雙曲線上一點(diǎn)P0(x0,y0)的切線方程是:利用上述方法可得圓錐曲線的切線方程如下:(1)過圓(x-a)證:設(shè)x有增量Δx,則對應(yīng)的u,y分別有增量Δu,Δy.因?yàn)樵邳c(diǎn)x處可導(dǎo),所以在點(diǎn)x處連續(xù).因此當(dāng)Δx→0時,Δu→0.當(dāng)Δu≠0時,由,且得:當(dāng)Δu=0時,公式也成立.

上面的證明其實(shí)不是一個很嚴(yán)格的證明,而且中間還會有不少的疑問,譬如,Δu=0時公式也成立,怎樣去理解;Δx→0時與Δu→0時的極限相等問題等等.因此同學(xué)們只要了解公式證明中的基本思想和方法即可,不必過多的去深究證明的過程.因?yàn)槭聦?shí)上,還有更嚴(yán)格的證明.證:設(shè)x有增量Δx,則對應(yīng)的u,y分別有增量Δu,Δy.因初一英語第二學(xué)期同步串講第七講

Doyouhaveaneraser?初一英語第二學(xué)期同步串講Doyouhaveasoccerball?復(fù)合函數(shù)求導(dǎo)教學(xué)課件soccerbasketballbadmintonbaseballballgamessoccerbasketballbadmintonbasetennis(racket)volleyballgolfPingpong(ball)tabletennistennis(racket)volleyballgolfPDoyouhaveanAmericanfootball/rugby?Yes,Ido./No,Idon’t.DoyouhaveanAmericanfootbaGolfclubDoyouhaveagolfball?Yes,Ido./No,Idon’t.GolfclubDoyouhaveagolfbaGroupworkNamesDoyouhavea…?BallsyouhaveLucyYes,Ido.basketballWhatballsdoyouhave?Ihave….GroupworkNamesDoyouhavea…1.你有一個排球嗎？是的，我有。2.你有表兄妹嗎?不,我沒有。3.她有網(wǎng)球拍嗎？是的，她有。7.咱們?nèi)ヌ咔虬?。Doyouhaveavolleyball?Yes,Ido.Doyouhaveanycousins?No,Idon’t.Doesshehaveatennisracket?Yes,shedoes.Let’splaysoccer.1.你有一個排球嗎？是的，我有。Doyou4.他有英英字典嗎？不，他沒有。他有一本英漢字典。5.他們有筆記本嗎？是的，他們有。6.讓我看一看。DoeshehaveanEnglishEnglishdictionary?No,hedoesn’t.HehasanEnglishChinesedictionary?Dotheyhaveanynotebooks?Yes,theydo.Let’shavealook.4.他有英英字典嗎？不，他沒有。他有一本英漢字典。DoeDoeshehaveatennisracket?Yes,hedoes./No,hedoesn't.Doeshehaveatennisracket?YDoesshehaveasoccerball?Yes,shedoes./No,shedoesn't.Doesshehaveasoccerball?YeDoeshehaveabasketball?Yes,hedoes.Doeshehaveabasketball?Yes,Dotheyhaveapuck?Yes,theydo./No,theydon’t.icehockeyDotheyhaveapuck?icehockeyA:Iwanttoplaybaseball.Doyouhaveabaseballbat?B:No,Idon’t.A:Doesyourbrotherhaveone?B:Yes,hedoes.Let’sgoandfindhim.A:Iwanttoplaybaseball.DoLet’splayvolleyball.Thatsoundsgood.Let’splayvolleyball.Let’splaybarbell.BarbellThatsoundsboring.Let’splaybarbell.BarbellThatLet’splaybowling.Thatsoundsinteresting.Let’splaybowling.ThatsoundsLet’splaybaseball.Thatsoundsdifficult.Let’splaybaseball.Let’sgoskateboarding.Thatsoundsfun.(exciting)Let’sgoskateboarding.Let’splayPingPong.Thatsoundsrelaxing.letsb.dosth.Let’shaveabreak.Let’sgoandplayPing-pong.Let’splayPingPong.letsb.dOurSchoolSportsCenterneedsnewmembers.Fillintheenrollmentformifyouwanttojointheclubs.Firstname__________Lastname__________Age________________Sex________________Class______________Club_______________Reason__________________________________What’syourfirstname,please?Howoldareyou?Whatclassareyouin?Whatclubdoyouwanttojoin?Why?(Because…Ilike…)OurSchoolSportsCenterneed寫出反義詞dislike______interesting______good_______well________easy________lost_________day________different_______big________first_________on________infrontof_________this_______these_____________bring______寫出反義詞Fillintheblanks1.A:Excuseme,_____you____abaseball?B:No,I_____.A:______Jim_____one?B:Yes,he______.Let’sgoandfindhim.2.A:Hi,Sally.B:Hi,Jane.A:Sally,________myfriend,Lucy.B:Hi,Lucy._____tomeetyou.C:Niceto_____you,_____.dohavedon’tDoeshavedoesthisisNicemeettooFillintheblanksdoA:Let’splayvolleyball.B:That______interesting,butI_____haveone.A:Well,_____youhaveatennisbat?B:Yes,I_____.Let’sgoandplaytennis.A:That’sagood______.soundsdon’tdodoideaA:Let’splayvolleyball.soundDoesBhaveaneraser?Istheshopopen?Whatdayistoday?Whatcantheydo?No,shedoesn’t.No,itisn’t.Itisclosed.TodayisFriday.Theygotothesupermarket.Part1DoesBhaveaneraser?No,sheWhatwouldBlike?HowmuchmoneydoesBhave?Whereisherpurse?Whatdoesshedo?

Sheonlyhastwoyuan.Herpurseisathome.SheborrowsfouryuanfromC.Part2She’dlikearedpencilandapencil-sharpener.WhatwouldBlike?Part2She’dDoesJimhaveadictionary?DoesAnnhaveadictionary?WhohasanEnglish-Chinesedictionary?4.WhencanLiLeigiveitback?

Part3No,hedoesn’t.Lucyhasone.HecangiveitbacknextMonday.No,shedoesn’t.DoesJimhaveadictionary?P

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