第二章導(dǎo)數(shù)及其應(yīng)用

1、第二章第二章 導(dǎo)數(shù)及其應(yīng)用導(dǎo)數(shù)及其應(yīng)用1. 導(dǎo)數(shù)的概念和意義3. 求導(dǎo)法則及基本導(dǎo)數(shù)表4. 反函數(shù)的求導(dǎo)法則5. 隱函數(shù)和參數(shù)函數(shù)求導(dǎo)6.無窮小量和微分2. 可導(dǎo)和連續(xù)的關(guān)系7. 高階導(dǎo)數(shù)導(dǎo)數(shù)的概念和意義導(dǎo)數(shù)的概念和意義AB0 xy0 x0 xxxy00()( )limlimxxyf xxf xxx 000( )()limxxf xf xxx0()fx過點(diǎn)00(,()xf x切線方程000( )()()()f xf xfxxx過點(diǎn)00(,()xf x法線方程0001( )()()()f xf xxxfx 例題例題( )() ( ),( )( ).f xxa g xg xafa設(shè)其中在點(diǎn) 處連續(xù)

2、，求1.( )( )( )limlim ( )( )xaxaf xf afag xg axa2.0(1)(1)( )lim1,2xffxf xx 設(shè)為可導(dǎo)函數(shù)，且滿足條件( )1(1)yf xf則曲線在點(diǎn)（，處的切線斜率為（ D ）A. 2 B. -1 C. D. -20(1)(1)lim12xffxx 由0(1)(1)(1)lim,(1)222xfxfffx 得得可導(dǎo)與連續(xù)的關(guān)系可導(dǎo)與連續(xù)的關(guān)系探討0yxx在處是否連續(xù)，是否可導(dǎo)。0yxx在處連續(xù)00( )(0)limlim10 xxxf xfxx 00( )(0)limlim10 xxxf xfxx 00( )(0)( )(0)limlim

3、00 xxf xff xfxx 連續(xù)不一定可導(dǎo)13yxy1sin,0 xxx0,0 x 可導(dǎo)與連續(xù)的關(guān)系可導(dǎo)與連續(xù)的關(guān)系0000()()lim()xf xxf xfxx 000()()()()f xxf xfxxx0()0 xx 當(dāng)時(shí)，000()()()()f xxf xx fxxx 000()()()()f xxf xx fxxx 000lim()()xf xxf x 可導(dǎo)一定連續(xù)導(dǎo)數(shù)的四則運(yùn)算法則導(dǎo)數(shù)的四則運(yùn)算法則( )( ) ( )( )f xg xfxg x( )( ) ( )( )( ) ( )f xg xfxg xf x g x2( )( ) ( )( ) ( )( ( )0)(

4、)( )f xfx g xf x g xg xg xgx導(dǎo)數(shù)的四則運(yùn)算法則導(dǎo)數(shù)的四則運(yùn)算法則 0()()( )( )( )( ) limxf xxg xxf xg xf xg xx 0()( )()( )limxf xxf xg xxg xx ( )( )fxg x0()()( ) ( )( )( ) limxf xxg xxf x g xf xg xx 0() ()( ) ()( ) ()( ) ( )limxf xx g xxf x g xxf x g xxf x g xx 0()()( )( )()( )limxg xxf xxf xf xg xxg xx ( ) ( )( ) ( )f

5、x g xf x g x導(dǎo)數(shù)的四則運(yùn)算法則導(dǎo)數(shù)的四則運(yùn)算法則0()( )( )()( )lim( )xf xxf xf xg xxg xg xx 01() ( )( ) ()lim() ( )xf xx g xf x g xxg xx g xx 21( ) ( )( ) ( )( )fx g xg x f xgx01() ( )( ) ( )( ) ( )( ) ()lim() ( )xf xx g xf x g xf x g xf x g xxg xx g xx 復(fù)合函數(shù)求導(dǎo)法則復(fù)合函數(shù)求導(dǎo)法則( )( )( ) .yf uuxyfx設(shè)與構(gòu)成復(fù)合函數(shù)( )( )( )uxxyf uux若在

6、處可導(dǎo)，在對(duì)應(yīng)點(diǎn)( )yfxx處可導(dǎo)，則復(fù)合函數(shù)在 可導(dǎo)且有( )( )xuxyyufuxdydy dudxdu dx反函數(shù)求導(dǎo)法反函數(shù)求導(dǎo)法( )yf x( )xg y反函數(shù)( )xg f x則x兩邊同時(shí)對(duì) 求導(dǎo)得11( )( ),( )( )gf xfxgf xfx從而( )yf x設(shè)函數(shù)在( , )a b 內(nèi)連續(xù)，嚴(yán)格單調(diào)且其值域?yàn)?( ,).( )( ,)A Bxg yA By又設(shè)其反函數(shù)在內(nèi)處有不為零的導(dǎo)數(shù)，00001( )()().()yf xxg ygf xfx則在對(duì)應(yīng)點(diǎn)處有導(dǎo)數(shù)，且基本導(dǎo)數(shù)公式基本導(dǎo)數(shù)公式( )0c1()xx(sin )cosxx(cos )sinxx 21(t

7、an )cosxx21(cot )sinxx 1(ln )xx()xxee21(arcsin )1xx21(arctan )1xx21(arccot )1xx 21(arccos )1xx ()lnxxaaa11(log)lnaxa x基本導(dǎo)數(shù)公式的推導(dǎo)基本導(dǎo)數(shù)公式的推導(dǎo)002cossinsin()sin22(sin )limlimxxxxxxxxxxxx 002sin2limlim cos()cos2xxxxxxx 002sinsincos()cos22(cos )limlimxxxxxxxxxxxx 002sin2limlim sin()sin2xxxxxxx 基本導(dǎo)數(shù)公式的推導(dǎo)基本導(dǎo)數(shù)公

8、式的推導(dǎo)2sin(sin )cossin (cos )(tan )()coscosxxxxxxxx2222cossin1coscosxxxx2cos(cos )sincos (sin )(cot )()sinsinxxxxxxxx2222sincos1sinsinxxxx 基本導(dǎo)數(shù)公式的推導(dǎo)基本導(dǎo)數(shù)公式的推導(dǎo)00(1)()limlimxxxxxxxxxeee eeexx lnln()()()xxaxaaeelnln( ln )lnlnxaxaxexaeaaa基本導(dǎo)數(shù)公式的推導(dǎo)基本導(dǎo)數(shù)公式的推導(dǎo)00ln(1)ln()ln(ln )limlimxxxxxxxxxx 0ln(1)1limxxxxxx

9、x ln11(log)()lnlnaxxaa x基本導(dǎo)數(shù)公式的推導(dǎo)基本導(dǎo)數(shù)公式的推導(dǎo)arcsinyx反函數(shù)sinxy導(dǎo)數(shù)cos y取倒數(shù)11coscosarcsinyx導(dǎo)數(shù)1x21x由圖可知211cosarcsin1xx基本導(dǎo)數(shù)公式的推導(dǎo)基本導(dǎo)數(shù)公式的推導(dǎo)反函數(shù)導(dǎo)數(shù)取倒數(shù)導(dǎo)數(shù)1x21x由圖可知arccosyxcosxysin y11sinsinar cosycx 211sinarccos1xx基本導(dǎo)數(shù)公式的推導(dǎo)基本導(dǎo)數(shù)公式的推導(dǎo)反函數(shù)導(dǎo)數(shù)取倒數(shù)導(dǎo)數(shù)1x21x由圖可知arctanyxtanxy21cos y22coscos (arctan )yx221cos (arctan )1xx基本導(dǎo)數(shù)公式

10、的推導(dǎo)基本導(dǎo)數(shù)公式的推導(dǎo)反函數(shù)導(dǎo)數(shù)取倒數(shù)導(dǎo)數(shù)1x21x由圖可知arccotyxcotxy21sin y22sinsin (arccot )yx 221sin (arccot )1xx例題例題 函數(shù)的求導(dǎo)函數(shù)的求導(dǎo)2sincoscos2xxyex求的導(dǎo)數(shù)2sin2coscos(sin) ( cos )2cos (2)xxxyexxx2sincoscossinsin2sin cos2cos2(ln2)2 cos2 cosxxxxxexxxxx2sincossinsin22(1cosln2)2 cosxxxexxx2231(3)xxyxx求的導(dǎo)數(shù)例題例題 函數(shù)的求導(dǎo)函數(shù)的求導(dǎo)兩邊先取絕對(duì)值后再取對(duì)數(shù)

11、得解：1ln2lnln 1ln 3ln 32yxxxxx兩邊對(duì) 求導(dǎo)得1211112(3)3yyxxxx229(1)2(9)xxxxx229(1)2(9)xxyyxxx練習(xí)練習(xí) 函數(shù)的求導(dǎo)函數(shù)的求導(dǎo)32(1)(2)(0)(1)()xx xxyxxex求的導(dǎo)數(shù)兩邊取對(duì)數(shù)得解：211111lnlnln(1)ln(2)ln(1)ln()33333xyxxxxexx兩邊同時(shí)對(duì) 求導(dǎo)得211 11121()3121xxxeyyxxxxex3221(1)(2) 11121()3(1)()121xxxx xxxeyxexxxxxex練習(xí)練習(xí) 函數(shù)的求導(dǎo)函數(shù)的求導(dǎo)( )(1)(2)(),(0)f xx xxx

12、nf設(shè)求00( )(0)(1)(2)()(0)limlim!0 xxf xfx xxxnfnxx隱函數(shù)的求導(dǎo)隱函數(shù)的求導(dǎo)2tan()325,dyxyx yxdx已知求解：x兩邊同時(shí)對(duì) 求導(dǎo)得：221(1)3(2)20cos ()yxyx yxy整理得222sec ()sec () 63 20 xyxyyxyx y2222sec ()6sec ()3xyxyyxyx隱函數(shù)的求導(dǎo)隱函數(shù)的求導(dǎo)0tan(),/xyxdyexyydx求解：x兩邊同時(shí)對(duì) 求導(dǎo)可得：2()sec ()()xyeyxyxyyxyy整理得：22sec ()sec ()xyxyyexeyyxyxxyyy22sec ()1sec

13、()xyxyy exyyxexxy 01,xy當(dāng)時(shí)，代入可得0/2xdydx練習(xí)練習(xí) 隱函數(shù)的求導(dǎo)隱函數(shù)的求導(dǎo)23xyex y求的導(dǎo)數(shù)x兩邊對(duì) 求導(dǎo)可得解：2()63xyeyxyxyx y移項(xiàng)并整理得263xyxyxyyeyxex再利用方程式得(2)(1)yxyyx xy練習(xí)練習(xí) 隱函數(shù)的求導(dǎo)隱函數(shù)的求導(dǎo)22arctanlnyxyx求的導(dǎo)數(shù)解：等式兩邊求導(dǎo)得2222222111 2221xyyxyyyxxyxyx2222xyyxyyxyxyxyyxyyxyyxy參數(shù)函數(shù)的求導(dǎo)參數(shù)函數(shù)的求導(dǎo)21xt arctanyt求22d ydx解：21/1/2dydy dttdxdx dtt22()()/d

14、ydydddtd ydxdxdxdxdx dt232 3(1 3 )4 (1)ttt練習(xí)練習(xí) 參數(shù)函數(shù)的求導(dǎo)參數(shù)函數(shù)的求導(dǎo)(sin )xa tt(1 cos )yat22dyd ydxdx求和/sinsin/cos1 cosdydy dtattdxdx dtaatt解：222(sin )(1 cos )sin (1 cos )()/(1 cos )/costtttdyddtd ytdxdxdx dtaat21(1 cos )at無窮小量和微分無窮小量和微分xa 時(shí)，( )0, ( )0f xg x( )( )( )( )f xg xf xg x和均為無窮小量( )( )f xg x無窮小量否？

15、不一定( )lim1,( )( )( )xaf xf xg xg x若與等價(jià)無窮小量.0 x 時(shí)，sin xxln(1) xx211 cos2xx無窮小量和微分無窮小量和微分( ),( )( )0 xxx都為無窮小量，更快( )x更高階無窮小量( )( )( )xxx( )( )xox( )lim,0( )xaxl lx同階無窮小量3300sinsintansincoslimlimxxxxxxxxx3300sincos sinsin (1 cos )limlimxxxxxxxxx230sin2sin12lim2xxxx無窮小量和微分無窮小量和微分00000()()limlim()xxf xxf xyfxxx 000()()()()f xxf xxfxx000()()()()f xxf xfxxxx 00()()()yf xxf xAxOx ？無窮小量和微分無窮小量和微分0( ),yf xxA設(shè)在 點(diǎn)附近有定義，假定有一個(gè)常數(shù)使得0( )yf xx則在 可微.:A xdfdy微分，記為或可微和可導(dǎo)等價(jià)00()()()(0)yf xxf xA xoxx 高階導(dǎo)數(shù)高階導(dǎo)數(shù)( )