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导数证明技巧大全：构造数、放缩法与同构策略
瀵兼暟璇佹槑棰樻€昏浣犲ご澶э紵鍏堟悶鎳傝繖涓変釜鏉€鎵嬮攺锛?/h3>
鍒氭帴瑙﹀鏁扮殑鏈嬪弸鍙兘浼氭兂锛氳繖鐜╂剰鍎夸笉灏辨槸绠楁枩鐜囧悧锛熶负浠€涔堣瘉鏄庨鑳介毦鍊掍竴鐗囦汉锛燄煒?浠婂ぉ鍜变滑灏辨幇寮€浜嗘弶纰庝簡璇粹€斺€斺€?strong>鈥嬫瀯閫犳暟銆佹斁缂╂硶銆佸悓鏋勭瓥鐣モ€?/strong>鈥嬭繖涓夋澘鏂э紝淇濆噯浣犵湅瀹屾暍瀵瑰鏁拌瘉鏄庨璇?灏辫繖锛?
涓€銆佹瀯閫犳暟锛氭妸澶ц薄瑁呭啺绠遍渶瑕佸嚑姝ワ紵
鈥?strong>鈥嬫牳蹇冮棶棰樷€?/strong>鈥嬶細鏋勯€犳暟鍚潃鐜勪箮锛屽埌搴曟槸涓暐鎿嶄綔锛?br/> 涓句釜渚嬪瓙浣犲氨鎳傦細瑕佽瘉鏄?褰搙>0鏃讹紝e耍 > 1+x"锛岀洿鎺ョ畻瀵兼暟鍙兘鍗″３銆傝繖鏃跺€欌€?strong>鈥嬫瀯閫犲樊鍑芥暟f(x)=e耍 -x-1鈥?/strong>鈥嬶紝鍐嶇爺绌跺畠鐨勫崟璋冩€э紝闂鐬棿鍙樼畝鍗曘€?/p>
鈥?strong>鈥嬪吀鍨嬬炕杞︾幇鍦衡€?/strong>鈥嬶細
鏈夊悓瀛︽妸f(x)=x虏-2x+1鍐欐垚(x-1)虏锛岀粨鏋滃彂鐜拌繖璐ф渶灏忓€兼槸0鈥斺€斾絾鍘熼瑕佽瘉鐨勬槸杩欎釜寮忓瓙姘歌繙鈮?鍟婏紒馃挕鐪嬪惂锛屾瀯閫犲浜嗗嚱鏁帮紝璇佹槑鐩存帴鐧界粰銆?/p>
鈥?strong>鈥嬮珮闃剁帺娉曗€?/strong>鈥嬶細
閬囧埌鍚弬鏁扮殑棰橈紝姣斿璇佹槑"褰揳>e鏃讹紝x虏 + a lnx >0"锛岃寰楀厛鈥?strong>鈥嬪垎绂诲弬鏁扳€?/strong>鈥嬨€傛妸寮忓瓙鏀规垚a > -x虏/lnx锛屽彸杈硅繖涓柊鍑芥暟鐨勬渶澶у€肩敤瀵兼暟涓€绠楋紝鑼冨洿灏卞嚭鏉ヤ簡銆?/p>
浜屻€佹斁缂╂硶锛氭暟瀛﹂噷鐨?缇庨鐩告満"鎬庝箞鐢紵
鈥?strong>鈥嬪繀鑳屽彛璇€鈥?/strong>鈥嬶細
- e耍 鈮+1锛坸=0鏃跺彇绛夛級
- lnx 鈮-1锛坸=1鏃跺彇绛夛級
- sinx 鈮锛坸鈮?鏃讹級
鈥?strong>鈥嬪疄鎴樻暀瀛︹€?/strong>鈥嬶細
瑕佽瘉"x - sinx >0锛坸>0锛?锛岀洿鎺ユ眰瀵煎お楹荤儲銆傚鐢╯inx 鈮杩欎釜鏀剧缉锛岀珛椹緱x - sinx 鈮?锛岀瓑鍙峰彧鍦▁=0鍑虹幇锛岃繖涓嶅氨鎼炲畾浜嗭紵
鈥?strong>鈥嬬炕杞﹂璀︹€?/strong>鈥嬶細
鍘诲勾妯¤€冩湁涓瑕佹眰璇佹槑"e耍 - x 鈮? + 0.5x虏"銆傛湁鑱槑铔嬬洿鎺ュ鐢╡耍 鈮?+x+0.5x虏锛岀粨鏋滈鐩璇佺殑鏄洿鏉剧殑涓嶇瓑寮忥紝鍙嶈€岃鎵ｅ垎鈥斺€旀斁缂╄繃澶翠篃涓嶈鍟婏紒馃槄
涓夈€佸悓鏋勭瓥鐣ワ細鎵惧璞′笉濡傛壘"缁撴瀯鍙岃優鑳?
鈥?strong>鈥嬬粡鍏告渚嬧€?/strong>鈥嬶細
纰板埌杩欑寮忓瓙锛?e耍 + x = 2 + ln(y+1)"锛岃€佹墜浼氫袱杈瑰悓鏃?1鍙樻垚e耍 +x +1 = 3 + ln(y+1)銆傜湅鍑烘潵浜嗗悧锛熷乏杈规槸e耍 + (x+1)锛屽彸杈规槸3 + ln(y+1)锛岃繖鏃跺€欎护t=x+1锛岀粨鏋勫氨瀵圭О浜嗐€?/p>
鈥?strong>鈥嬪繀鏉€鎶€琛ㄢ€?/strong>鈥嬶細
| 鍘熷紡鐗瑰緛 | 鍚屾瀯鍙樺舰濂楄矾 | 閫傜敤鍦烘櫙 |
|---|---|---|
| 鍚玡耍鍜宭nx | 鏀瑰啓鎴恊^{lnx}鎴杔ne耍 | 鎸囨暟瀵规暟娣峰悎棰?/td> |
| 澶氶」寮?鍒嗗紡 | 鎻愬彇鍏洜寮忔垨閫氬垎 | 澶嶆潅鍒嗗紡璇佹槑 |
| 鍚笁瑙掑嚱鏁?/td> | 鐢╯in虏x+cos虏x=1杞崲 | 涓夎鎭掔瓑寮忕浉鍏宠瘉鏄?/td> |
鈥?strong>鈥嬬嫭瀹惰瑙ｂ€?/strong>鈥嬶細
鏈€杩戜笁骞撮珮鑰冨鏁板帇杞撮锛岃秴杩?0%閮界敤鍒颁簡鍚屾瀯鎬濇兂銆傚幓骞撮偅閬撳叏缃戝枈闅剧殑"鏋佸€肩偣鍋忕Щ"棰橈紝鐢ㄥ悓鏋勬硶5姝ュ氨鑳界牬瑙ｏ紝姣斾紶缁熸柟娉曞揩涓€鍊嶃€?/p>
鍥涖€佺粍鍚堟嫵鎬庝箞鎵擄紵鐪嬭繖涓浠欐搷浣?/h3>
閬囧埌"璇佹槑锛氬綋x>0鏃讹紝e耍 - ln(x+1) >1"锛屽挶浠彲浠ワ細
- 鈥?strong>鈥嬫瀯閫犫€?/strong>鈥媐(x)=e耍 - ln(x+1) -1
- 鈥?strong>鈥嬫斁缂┾€?/strong>鈥媗n(x+1) 鈮
- 寰楀埌f(x) 鈮耍 -x -1
- 鍐嶅埄鐢ㄥ凡鐭ョ粨璁篹耍 鈮+1
- 鍙岄噸鏀剧缉閿佸畾鑳滃眬
浜斻€佹柊鎵嬫渶璇ヨ翰鐨勪笁澶у潙
- 鈥?strong>鈥嬩贡鐢ㄦ礇蹇呰揪鈥?/strong>鈥嬶細鏄庢槑棰樼洰璇?鐢ㄥ鏁拌瘉鏄?锛屼綘闈炴嬁楂樻暟宸ュ叿瑙ｉ锛屽啓寰楀啀瀵逛篃闆跺垎锛?/li>
- 鈥?strong>鈥嬪拷瑙嗗畾涔夊煙鈥?/strong>鈥嬶細鍦╨nx鐨勯閲岃璁簒鈮?鐨勬儏鍐碉紝鐩稿綋浜庡湪娓告吵姹犻噷鎵鹃波楸尖€斺€旂櫧璐瑰姴
- 鈥?strong>鈥嬫毚鍔涙眰瀵间笂澶粹€?/strong>鈥嬶細鏈変釜棰樻眰浜嗕笁娆″杩樻病缁撴灉锛屽叾瀹炵敤鍚屾瀯娉曟棭璇ョ湅鐮寸巹鏈轰簡
鏈€鍚庤鐐瑰ぇ瀹炶瘽
杩欎簺鎶€宸у氨鍍忔暟瀛﹂噷鐨?濂楄矾"锛屼絾鍒互涓鸿儗浜嗗叕寮忓氨鑳芥í琛岃€冨満銆傚幓骞存湁涓皟鏌ユ樉绀猴紝鑳界伒娲昏繍鐢ㄨ繖涓夌鏂规硶鐨勫鐢燂紝瀵兼暟棰樺緱鍒嗙巼鏄璁扮‖鑳岃€呯殑3鍊嶃€備笅娆″仛棰樺墠鍏堥棶鑷繁锛氳繖棰樿兘鐢ㄦ瀯閫?鏀剧缉/鍚屾瀯鐨勬€濊矾鐮村悧锛熷杩欐牱璁粌锛屼綘绂绘弧鍒嗗氨涓嶈繙鍟︼紒