反函数的导数公式 反函数求导法则
\u53cd\u51fd\u6570\u7684\u5bfc\u6570\u4e0e\u539f\u51fd\u6570\u7684\u5bfc\u6570\u6709\u4ec0\u4e48\u5173\u7cfb\u539f\u51fd\u6570\u7684\u5bfc\u6570\u7b49\u4e8e\u53cd\u51fd\u6570\u5bfc\u6570\u7684\u5012\u6570\u3002
\u8bbey=f(x),\u5176\u53cd\u51fd\u6570\u4e3ax=g(y),
\u53ef\u4ee5\u5f97\u5230\u5fae\u5206\u5173\u7cfb\u5f0f\uff1ady=(df/dx)dx ,dx=(dg/dy)dy .
\u90a3\u4e48,\u7531\u5bfc\u6570\u548c\u5fae\u5206\u7684\u5173\u7cfb\u6211\u4eec\u5f97\u5230,
\u539f\u51fd\u6570\u7684\u5bfc\u6570\u662f df/dx = dy/dx,
\u53cd\u51fd\u6570\u7684\u5bfc\u6570\u662f dg/dy = dx/dy .
\u6240\u4ee5,\u53ef\u4ee5\u5f97\u5230 df/dx = 1/(dg/dx) .
\u6269\u5c55\u8d44\u6599\uff1a
\u53cd\u51fd\u6570\u5b58\u5728\u5b9a\u7406
\u5b9a\u7406\uff1a\u4e25\u683c\u5355\u8c03\u51fd\u6570\u5fc5\u5b9a\u6709\u4e25\u683c\u5355\u8c03\u7684\u53cd\u51fd\u6570\uff0c\u5e76\u4e14\u4e8c\u8005\u5355\u8c03\u6027\u76f8\u540c\u3002
\u5728\u8bc1\u660e\u8fd9\u4e2a\u5b9a\u7406\u4e4b\u524d\u5148\u4ecb\u7ecd\u51fd\u6570\u7684\u4e25\u683c\u5355\u8c03\u6027\u3002
\u8bbey=f(x)\u7684\u5b9a\u4e49\u57df\u4e3aD\uff0c\u503c\u57df\u4e3af(D)\u3002\u5982\u679c\u5bf9D\u4e2d\u4efb\u610f\u4e24\u70b9x1\u548cx2\uff0c\u5f53x1y2\uff0c\u5219\u79f0y=f(x)\u5728D\u4e0a\u4e25\u683c\u5355\u8c03\u9012\u51cf\u3002
\u8bc1\u660e\uff1a\u8bbef\u5728D\u4e0a\u4e25\u683c\u5355\u589e\uff0c\u5bf9\u4efb\u4e00y\u2208f(D)\uff0c\u6709x\u2208D\u4f7ff(x)=y\u3002
\u800c\u7531\u4e8ef\u7684\u4e25\u683c\u5355\u589e\u6027\uff0c\u5bf9D\u4e2d\u4efb\u4e00x'x\uff0c\u90fd\u6709y''>y\u3002\u603b\u4e4b\u80fd\u4f7ff(x)=y\u7684x\u53ea\u6709\u4e00\u4e2a\uff0c\u6839\u636e\u53cd\u51fd\u6570\u7684\u5b9a\u4e49\uff0cf\u5b58\u5728\u53cd\u51fd\u6570f-1\u3002
\u4efb\u53d6f(D)\u4e2d\u7684\u4e24\u70b9y1\u548cy2\uff0c\u8bbey1<y2\u3002\u800c\u56e0\u4e3af\u5b58\u5728\u53cd\u51fd\u6570f-1\uff0c\u6240\u4ee5\u6709x1=f-1(y1)\uff0cx2=f-1(y2)\uff0c\u4e14x1\u3001x2\u2208D\u3002
\u82e5\u6b64\u65f6x1\u2265x2\uff0c\u6839\u636ef\u7684\u4e25\u683c\u5355\u589e\u6027\uff0c\u6709y1\u2265y2\uff0c\u8fd9\u548c\u6211\u4eec\u5047\u8bbe\u7684y1<y2\u77db\u76fe\u3002
\u56e0\u6b64x1<x2\uff0c\u5373\u5f53y1<y2\u65f6\uff0c\u6709f-1(y1)<f-1(y2)\u3002\u8fd9\u5c31\u8bc1\u660e\u4e86\u53cd\u51fd\u6570f-1\u4e5f\u662f\u4e25\u683c\u5355\u589e\u7684\u3002
\u5982\u679cf\u5728D\u4e0a\u4e25\u683c\u5355\u51cf\uff0c\u8bc1\u660e\u7c7b\u4f3c\u3002
\u53c2\u8003\u8d44\u6599\uff1a
\u53cd\u51fd\u6570_\u767e\u5ea6\u767e\u79d1
\u53cd\u51fd\u6570\u7684\u6c42\u5bfc\u6cd5\u5219\u662f\uff1a\u53cd\u51fd\u6570\u7684\u5bfc\u6570\u662f\u539f\u51fd\u6570\u5bfc\u6570\u7684\u5012\u6570\u3002\u4f8b\u9898\uff1a\u6c42y=arcsinx\u7684\u5bfc\u51fd\u6570\u3002 \u9996\u5148\uff0c\u51fd\u6570y=arcsinx\u7684\u53cd\u51fd\u6570\u4e3ax=siny\uff0c\u6240\u4ee5\uff1ay\u2018=1/sin\u2019y=1/cosy
\u56e0\u4e3ax=siny\uff0c\u6240\u4ee5cosy=\u221a1-x2
\u6240\u4ee5y\u2018=1/\u221a1-x2\u3002
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\u6269\u5c55\u8d44\u6599\uff1a
\u4e00\u822c\u5730\uff0c\u8bbe\u51fd\u6570y=f(x)(x\u2208A)\u7684\u503c\u57df\u662fC\uff0c\u6839\u636e\u8fd9\u4e2a\u51fd\u6570\u4e2dx,y \u7684\u5173\u7cfb\uff0c\u7528y\u628ax\u8868\u793a\u51fa\uff0c\u5f97\u5230x= g(y). \u82e5\u5bf9\u4e8ey\u5728C\u53cd\u51fd\u6570\u4e2d\u7684\u4efb\u4f55\u4e00\u4e2a\u503c\uff0c\u901a\u8fc7x= g(y)\uff0cx\u5728A\u4e2d\u90fd\u6709\u552f\u4e00\u7684\u503c\u548c\u5b83\u5bf9\u5e94\uff0c\u90a3\u4e48\uff0cx= g(y)\u5c31\u8868\u793ay\u662f\u81ea\u53d8\u91cf\uff0cx\u662f\u56e0\u53d8\u91cf\u662fy\u7684\u51fd\u6570\uff0c\u8fd9\u6837\u7684\u51fd\u6570x= g(y)(y\u2208C)\u53eb\u505a\u51fd\u6570y=f(x)(x\u2208A)\u7684\u53cd\u51fd\u6570\uff0c\u8bb0\u4f5cy=f^(-1) (x) \u53cd\u51fd\u6570y=f^(-1) (x)\u7684\u5b9a\u4e49\u57df\u3001\u503c\u57df\u5206\u522b\u662f\u51fd\u6570y=f(x)\u7684\u503c\u57df\u3001\u5b9a\u4e49\u57df\u3002
大部分偶函数不存在反函数（当函数y=f(x)，定义域是{0}且f(x)=C（其中C是常数），则函数f(x)是偶函数且有反函数，其反函数的定义域是{C}，值域为{0}）。奇函数不一定存在反函数，被与y轴垂直的直线截时能过2个及以上点即没有反函数。若一个奇函数存在反函数，则它的反函数也是奇函数。
绛旓細浣犵殑鐞嗚В鏈夌偣闂,鈥鍙嶅嚱鏁扮殑瀵兼暟绛変簬鐩存帴鍑芥暟瀵兼暟鐨勫掓暟鈥濈殑鎰忔濇槸:浠=g(y)鏄痽=f(x)鐨勫弽鍑芥暟,鍒:g'(y)=1/f'(x)灏辨嬁浣犵殑渚嬪瓙鏉ヨ鏄巠=x^2(涓嶅Θ璁緓鈮0)鐨勫弽鍑芥暟鏄:x=鈭歽涓轰簡琛ㄨ堪涓婄殑涔犳儻鎬,鎴戜滑涓鑸浠栫殑鍙嶅嚱鏁版槸:y=鈭歺浣嗘槸鍦ㄦ眰瀵兼暟鐨勬椂鍊欏氨涓嶈兘杩欐牱浜嗗簲璇ユ槸杩欐牱:f(x)=x^2鐨勫弽鍑芥暟涓:...
绛旓細瀹炰緥锛 璁＄畻鍑芥暟 f(x) = x^3 - 3x^2 鐨鍙嶅嚱鏁 f^(-1)(x) 鐨勫鏁般傞鍏堬紝鎴戜滑闇瑕佹壘鍒板師鍑芥暟鐨勫弽鍑芥暟銆傚亣璁惧弽鍑芥暟涓 y = f^(-1)(x)锛岄偅涔堟垜浠彲浠ヨ x = y^3 - 3y^2銆備负浜嗘眰瀵硷紝鎴戜滑灏 x 瑙嗕负 y 鐨勫嚱鏁帮紝鍗 x = g(y)锛岀劧鍚庝唬鍏ュ師鍑芥暟鐨勫鏁板叕寮锛歞g(y)/dy = 3y^2 ...
绛旓細9銆侊紙fg)'=f'g+fg'锛屽嵆绉殑瀵兼暟绛変簬鍚勫洜寮忕殑瀵兼暟涓庡叾瀹冨嚱鏁扮殑绉紝鍐嶆眰鍜屻10銆侊紙f/g)'=(f'g-fg')/g^2锛屽嵆鍟嗙殑瀵兼暟锛屽彇闄ゅ嚱鏁扮殑骞虫柟涓洪櫎寮忋傝闄ゅ嚱鏁扮殑瀵兼暟涓庨櫎鍑芥暟鐨勭Н鍑忓幓琚櫎鍑芥暟涓庨櫎鍑芥暟鐨勫鏁扮殑绉殑宸负琚櫎寮忋11銆侊紙f^(-1)(x))'=1/f'(y)锛屽嵆鍙嶅嚱鏁扮殑瀵兼暟鏄師鍑芥暟瀵兼暟...
绛旓細鍙嶅嚱鏁扮殑瀵兼暟鍙互閫氳繃鍘熷嚱鏁扮殑瀵兼暟姹傚緱銆傚叿浣撴潵璇达紝濡傛灉涓涓嚱鏁癴(x)鍦ㄥ叾瀹氫箟鍩熷唴鍙锛屼笖鍏跺鏁颁笉涓洪浂锛岄偅涔坒(x)鐨勫弽鍑芥暟g(x)鍦ㄥ搴旂殑瀹氫箟鍩熷唴涔熷彲瀵硷紝涓攇'(x) = 1/f'(g(x))銆傝鐞嗚В杩欎釜鍏紡锛岄鍏堥渶瑕佷簡瑙ｅ弽鍑芥暟鐨勫畾涔夈傚弽鍑芥暟鏄竴绉嶅皢鍘熷嚱鏁扮殑杈撳嚭浣滀负杈撳叆锛岃緭鍑哄師鍑芥暟鐨勮緭鍏ョ殑鍑芥暟銆...
绛旓細鍙嶅嚱鏁扮殑姹傚娉曞垯鏄細鍙嶅嚱鏁扮殑瀵兼暟鏄師鍑芥暟瀵兼暟鐨勫掓暟銆備緥棰橈細姹倅=arcsinx鐨勫鍑芥暟銆 棣栧厛锛屽嚱鏁皔=arcsinx鐨勫弽鍑芥暟涓簒=siny锛屾墍浠ワ細y鈥=1/sin鈥檡=1/cosy 鍥犱负x=siny锛屾墍浠osy=鈭1-x2 鎵浠鈥=1/鈭1-x2銆傚悓鐞嗗彲浠ユ眰鍏朵粬鍑犱釜鍙嶄笁瑙掑嚱鏁扮殑瀵兼暟銆傛墍浠ヤ互鍚庡湪姹傛秹鍙婂埌鍙嶅嚱鏁扮殑瀵兼暟鏃讹紝鍏堝皢鍙...
绛旓細杩欎釜缁撹鍙互绠鍗曡〃杈句负锛鍙嶅嚱鏁扮殑瀵兼暟绛変簬鐩存帴鍑芥暟瀵兼暟鐨勫掓暟銆備緥锛氳x=siny锛寉鈭圼−蟺2,蟺2]涓虹洿鎺ュ鏁帮紝鍒檡=arcsinx鏄畠鐨勫弽鍑芥暟锛屾眰鍙嶅嚱鏁扮殑瀵兼暟銆傝В锛氬嚱鏁皒=siny鍦ㄥ尯闂村唴鍗曡皟鍙锛宖鈥(y)=cosy鈮0 鍥犳锛岀敱鍏紡寰 (arcsinx)鈥=1(siny)鈥=1cosy=11−sin2y−−...
绛旓細arctanx鐨勫鏁版槸1/(1+x^2)銆備负浜嗘帹瀵糰rctanx鐨勫鏁帮紝鎴戜滑鍙互浣跨敤鍙嶅嚱鏁扮殑瀵兼暟鍏紡銆傞鍏堬紝鎴戜滑鐭ラ亾tanx鐨勫鏁版槸sec^2x锛屽嵆(1+tan^2x)銆傜敱浜巃rctanx鏄痶anx鐨勫弽鍑芥暟锛屾牴鎹弽鍑芥暟鐨勫鏁板叕寮忥紝鎴戜滑鏈夛細濡傛灉y = f(x) 鐨勫鏁颁负 f'(x)锛岄偅涔堝叾鍙嶅嚱鏁 x = g(y) 鐨勫鏁颁负 1/f'(y)銆傚皢tanx...
绛旓細鍙嶅嚱鏁扮殑楂橀樁瀵兼暟鐨勮绠楁柟娉曞彲浠ラ氳繃鍙嶅嚱鏁扮殑姹傚娉曞垯鍜屽鍚堝嚱鏁扮殑姹傚娉曞垯杩涜璁＄畻銆備竴銆佸弽鍑芥暟鐨勬眰瀵煎彂鍒 鍙嶅嚱鏁扮殑姹傚娉曞垯鏄細鍙嶅嚱鏁扮殑瀵兼暟鏄師鍑芥暟瀵兼暟鐨勫掓暟銆傚嵆濡傛灉鍘熷嚱鏁 y=f(x) 鐨勫鏁颁负 f鈥(x)锛岄偅涔堝弽鍑芥暟 x=g(y) 鐨勫鏁 g鈥(y) 绛変簬 f鈥(x)/y鈥=1/y鈥层傝繖鏄洜涓哄弽鍑芥暟涓庡師...
绛旓細鍙嶅嚱鏁版眰瀵煎叕寮忕殑鍘熺悊鏄熀浜庡嚱鏁扮殑鍙嶅嚱鏁颁笌鍏跺師鍑芥暟涔嬮棿鐨勫叧绯讳互鍙婂鏁扮殑瀹氫箟銆傚弽鍑芥暟鐨勫鏁扮瓑浜庡師鍑芥暟瀵兼暟鐨勫掓暟涔樹互鍙嶅嚱鏁扮殑寰垎鍏冪殑鍙樺寲鐜囥傝繖鏍烽氳繃鍒╃敤鍙嶅嚱鏁扮殑瀹氫箟鍙婂叾鎬ц川锛屽苟缁撳悎寰Н鍒嗗師鐞嗭紝灏辫兘寰楀埌鍙嶅嚱鏁扮殑瀵兼暟鍏紡銆傚弽鍑芥暟姹傚鍏紡鐨勫簲鐢ㄥ墠鎻愭槸瀵瑰弽鍑芥暟鍜屽叾鍘熷嚱鏁版湁涓瀹氱悊瑙ｃ傚綋鎴戜滑瀵瑰師鍑芥暟瀹氫箟鍩...
绛旓細鍙嶅嚱鏁浜岄樁瀵兼暟鍏紡鏄痽''=-y'*d²x/dy²銆備簩闃跺鏁帮紝鏄師鍑芥暟瀵兼暟鐨勫鏁锛屽皢鍘熷嚱鏁拌繘琛屼簩娆℃眰瀵笺備竴鑸殑锛屽嚱鏁皔=f锛坸锛夌殑瀵兼暟y'=f'锛坸锛変粛鐒舵槸x鐨勫嚱鏁帮紝鍒檡''=f''锛坸锛夌殑瀵兼暟鍙仛鍑芥暟y=f锛坸锛夌殑浜岄樁瀵兼暟銆備竴鑸潵璇达紝璁惧嚱鏁皔=f锛坸锛夛紙x鈭圓锛夌殑鍊煎煙鏄疌锛岃嫢鎵惧緱鍒颁竴涓...
