[經(jīng)濟(jì)學(xué)]第12講求導(dǎo)法則(存儲(chǔ)版)
【正文】 ??? ? xe xxe 55 ???例 19 )0( , 1)||ln( ??? xxx證明：證 ????????0 , )l n (0 , ln||lnxxxxxyxxxx1)( l n) ||ln ( , 0 ????? 時(shí)當(dāng)xxxxxx1)(1))( l n () ||ln ( , 0 ?????????? 時(shí)當(dāng)綜上所述 , . )0( , 1)||ln( ??? xxx例 20 . , )l n ( 22 yaxxy ???? 求)(l n ( 22 ????? axxy)1(12222 axxaxx ?????)(1 2222axxaxx??????221ax ??解 1) )l n (( 2222axaxx?????例 21 . ,2 2s i n yy x ?? 求? ???? 2s in2 xy)(s i n2ln2 2s in 2 ??? xx)(c os2ln2 22s in 2 ??? xxxxxx 2c os2ln2 2s in 2 ???解 )0 ( ln)( ??? aaaa xx例 22 1a r c t a n )( ???xy.1 1 2x??? )1(1 222xxx ????) () ( 1 1112????xx解 例 23 1a r c ta nyyx?? ，求. ,2c o t yxy ?? 求)(2)2c s c(2c o t21)2( c o t2c o t21 2 ????????xxxxxy21)2c s c(2c o t21 2????xx2t a n2c s c41 2 xx ??解 例 24 按復(fù)合函數(shù)求導(dǎo)法則 ) 1ln ( 2 ???? xy) 1|| ( 12 ??? xx x12212 ??? xx)( )1( ln21 2 ??? x解 注意利用函數(shù) 的性質(zhì) 例 25 2 l n 1 , | | 1 , .y x x y ?? ? ?設(shè)求.??????? ?????? xxxxy 11lnc o s11lnc o s2??????? ???????? ?????? xxxxxx 11ln11lns i n11lnc o s2))39。求 39。求 , 1c o ss i n2 yxxxy ?????)( 2 ??? xy )( si n ?? x )( c o s ?? x )1( ??例 5 ，設(shè) nnnnn axaxaxaxay ?????? ??? 121110 ?)()()()()( 122110 ???????????? ??? nnnnn axaxaxaxay ?10 ?? nxna。求 , l o g yxy a ??解 ????????axlnln)(l nln 1 ?? xaax ln1? ln1)( log axxa ??例 9 已知 )3()2)(1()3)(2()1( ????????? xxxxxxy)2)(1()3)(1()3)(2( ????????? xxxxxx2)23)(13()33)(13()33)(23(| 3 ??????????? ?xy故。1l n()1(l n(11ln2s i n xxxx ?????????????解 例 26 2 1 c o s l n , ( 1 , 1 ) , 1 xy x yx??? ?? ? ??? ???設(shè)求????????????????????xxxx111111ln2s i n??????????xxx 11ln2s i n122 并不難 設(shè) y = f (x) 可導(dǎo) , 則 ??))(( s i n xf )()(c o s xfxf ????))( s i n( xf xxf c o s)( s i n ????))(( l n xf)()(xfxf ? )0)(( ?xf??))( l n( xfxxf 1)(l n ????)( )( xfe )()( xfe xf ????))(( xef xx eef ?? )(證明：在 (–a, a)內(nèi)可導(dǎo)的奇函數(shù)的導(dǎo)數(shù)是偶函數(shù)；偶函數(shù)的導(dǎo)數(shù)是奇函數(shù)。求 yxxy ????? , )11( a r c s i n解 sin 在 y x = 2 ,2 )(???yy c os1)(s i n1 ??? 你覺(jué)得做完了嗎？ 例 15 而 于是 22 1s i n1c os xyy ????211c o s1)(arcs i nxyxy??????)11( ??? x 1)1( 11)( a r c s i n 2?????? xxx。( 011 ???? ? xxx 自變量對(duì)其本身的導(dǎo)數(shù)為 1 )(1dd 1 ???????? ?xxx 211)1( ??? ???? xx ,12x??.3)( 23 xx ??)()( 21 ??? xx 211212121 ?? ?? xx ,21x?例 1 3. 指數(shù)函數(shù) xaaxy xxxxx ????? ?????? 00l i ml i mQ ?
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