函数f(xy)对x求偏导数是什么 z是xy的隐函数 F（x，y，z）对x求偏导数和F'x这两个...
\u4e00\u4e2a\u51fd\u6570f\uff08x\uff0cy\uff09\u4ec0\u4e48\u65f6\u5019\u5bf9x\u6c42\u504f\u5bfc\u548c\u5bf9y\u6c42\u504f\u5bfc\u76f8\u7b49\u5462\uff1f\u8fd9\u4e2a\u6709\u4ec0\u4e48\u7ed3\u8bba\u5417\uff1ff\uff08xy\uff09\u53ef\u4ee5\u5417\uff1ff''xy=f''yx\uff0c\u8fd9\u662f\u53ef\u4ee5\u786e\u5b9a\u7684
\u89e3\uff1a\u5bf9\u4e8e\u8fd9\u4e00\u9053\u9898\uff0c\u4e0d\u59a8\u8bbeu=F\uff08x\uff0cy\uff0cz\uff09\uff0cz=z\uff08x\uff0cy\uff09\u90a3\u4e48u\u5bf9\u4e8ex\u7684\u504f\u5bfc\u6570\u7b49\u4e8e\uff1a
∂u/∂x =∂F/∂x+∂F/∂z*∂z/∂x\uff0c
\u8fd9\u662f\u6b63\u786e\u7684\uff0c
\u540c\u65f6\u4e5f\u53ef\u4ee5\u5199\u6210
∂u/∂x = F'x + F'z * z\u2018\uff0c
\u5176\u4e2dF'x\u8868\u793a\u5bf9\u4e8e\u51fd\u6570u\u4e2d\u7684\u81ea\u53d8\u91cfx\u6c42\u5bfc\uff0c\u4e5f\u5c31\u662f\u4e0d\u628az=z\uff08x\uff0cy\uff09\u4ee3\u5165\u51fd\u6570F\uff08x\uff0cy\uff0cz\uff09\u65f6\uff0cu\u6216\u8005\u8bf4F\uff08x\uff0cy\uff0cz\uff09\u5bf9\u4e8ex\u6c42\u5bfc\u7684\u5bfc\u6570\u503c\uff0c\u4e5f\u5c31\u662f\u8bf4\uff0c\u6b64\u65f6\u8ba1\u7b97F'x\u65f6\u662f\u628ax\uff0cy\uff0cz\u90fd\u770b\u6210\u662f\u5404\u81ea\u72ec\u7acb\u7684\u81ea\u53d8\u91cf\uff1b\u540c\u6837\u7684\u9053\u7406\u3002\u5f0f\u5b50\u4e2dF'z * z\u2018\u8868\u793au\u6216\u8005\u8bf4F\uff08x\uff0cy\uff0cz\uff09\u5148\u5bf9\u4e8ez\u6c42\u5bfc\uff0c\u800cz\u53c8\u662fx\u7684\u51fd\u6570\uff0c\u6545\u7531\u6c42\u5bfc\u5f97\u94fe\u5f0f\u6cd5\u5219\u53ef\u77e5\u8fd8\u9700\u518d\u4e58\u4ee5z\u5bf9\u4e8ex\u7684\u5bfc\u6570\u3002
\u9700\u8981\u6ce8\u610f\u7684\u662f\uff1a\u5bf9\u4e8e\u5177\u4f53\u7684\u51fd\u6570\u6765\u8bf4\uff0c\u82e5\u5df2\u7ecf\u628az=z\uff08x\uff0cy\uff09\u4ee3\u5165\u51fd\u6570u\u4e2d\u6216\u8005\u8bf4F\uff08x\uff0cy\uff0cz\uff09\u4e2d\uff0c\u4e5f\u5c31\u662f\u53d8\u6210\u4e86F\uff08x\uff0cy\uff0cz\uff08x\uff0cy\uff09\uff09\uff0c\u6b64\u65f6\u81ea\u53d8\u91cfz\u5df2\u7ecf\u4e0d\u5b58\u5728\u4e86\uff0c\u5373F'z=0\uff0c\u4ea6\u5373\u4e0a\u5f0f\u4e2d F'z * z\u2018=0\uff0c\u6b64\u65f6\u663e\u7136\u6709
∂u/∂x = F\u2019x + 0\uff0c
\u7efc\u4e0a\u53ef\u4ee5\u5c06\u4e0a\u8ff0\u4e24\u79cd\u60c5\u51b5\u5408\u5e76\u5199\u6210
∂u/∂x = F'x + F'z * z\u2018\uff0c
\u53ea\u662f\u8981\u8bb0\u4f4f\uff0c\u8fd9\u53ea\u662f\u5f62\u5f0f\u4e0a\u7684\u8868\u793a\uff0c\u5728\u5177\u4f53\u7b97\u7684\u65f6\u5019\uff0c\u8981\u4e48\u5148\u5c06z=z\uff08x\uff0cy\uff09\u4ee3\u5165F(x\uff0cy\uff0cz)\u4e2d\u540e\u8fdb\u884c\u6c42\u5bfc\uff0c\u8981\u4e48\u5148\u4fdd\u7559z\u7684\u5f62\u5f0f\uff08\u6b64\u65f6\u8ba1\u7b97F\u2018x\u548cF'z\u65f6\u628ax\uff0cy\uff0cz\u90fd\u770b\u6210\u662f\u5404\u81ea\u72ec\u7acb\u7684\u81ea\u53d8\u91cf\uff09\u6309\u4e0a\u5f0f\u8fdb\u884c\u6c42\u5bfc\uff0c\u4e4b\u540e\u518d\u5c06z=z\uff08x\uff0cy\uff09\u4ee3\u5165\u6240\u6c42\u5f97\u7684\u5f0f\u5b50\u4e2d\uff0c\u4ece\u800c\u6d88\u53bbz\u800c\u4e3a\u53ea\u5173\u4e8ex\uff0cy\u7684\u51fd\u6570\u3002 \u5f53\u7136\u53ea\u8981\u8ba1\u7b97\u6b63\u786e\uff0c\u6700\u540e\u7ed3\u679c\u80af\u5b9a\u662f\u4e00\u6837\u7684\u3002
如图所示：
绛旓細瀵箈鐨勫亸瀵兼暟 =f'(xy) 脳y =yf'(xy)
绛旓細瀵箈鐨勫亸瀵兼暟 =f'(xy) 脳y =yf'(xy)
绛旓細x=y4銆佸洜姝わ紝鍑芥暟f锛x锛寉锛=xy瀵箈鐨勫亸瀵兼暟鏄y銆5銆佺敤鏁板鍏紡锛屾垜浠彲浠ヨ〃绀轰负锛?锛坸y锛/?x=y
绛旓細绛旀鏄 f鈥橈紙xy锛*y銆傝繖涓槸澶嶅悎娣峰悎瀵兼暟銆備护 u=xy锛屽厛瀵 f锛坲锛夋眰瀵兼暟锛岀劧鍚庡湪瀵箄=xy 姹傚鏁 浣跨敤鐨勭殑鏄摼寮忔硶鍒
绛旓細濡傚浘鎵绀猴細
绛旓細鍋忓鏁鍏紡鏄細1銆亁鏂瑰悜鐨勫亸瀵 璁炬湁浜屽厓鍑芥暟z=f(x,y)锛岀偣锛坸0,y0)鏄叾瀹氫箟鍩烡鍐呬竴鐐广傛妸y鍥哄畾鍦▂0鑰岃x鍦▁0鏈夊閲忊柍x锛岀浉搴斿湴鍑芥暟z=f(x,y)鏈夊閲忥紙绉颁负瀵箈鐨锛囧亸澧為噺锛夆柍z=f(x0+鈻硏,y0)-f(x0,y0)銆傚鏋溾柍z涓庘柍x涔嬫瘮褰撯柍x鈫0鏃剁殑鏋侀檺瀛樺湪锛岄偅涔堟鏋侀檺鍊肩О涓哄嚱鏁皕=f(x,y...
绛旓細瑕佷細姹傚氨濂斤紝姣斿鍑芥暟锛歠(x,y)=sinx+xy 姹傚亸瀵兼暟鐨勬柟娉曞氨鏄鍏舵眰鍏ㄥ鏁帮紝鍗筹細df(x,y)=cosxdx+ydx+xdy=(cosx+y)dx+xdy 鍒欑瓑鍙峰悗闈x鍓嶉潰鐨勭郴鏁颁负鍑芥暟瀵箈鐨勫亸瀵兼暟锛宒y鍓嶉潰鐨勭郴鏁板嵆涓哄y鐨勫亸瀵兼暟锛屾墍浠ワ細鍑芥暟f(x,y)瀵箈鐨勫亸瀵兼暟=cosx+y;鍑芥暟f(x,y)瀵箉鐨勫亸瀵兼暟=x....
绛旓細鐢卞亸瀵兼暟瀹氫箟锛屽彲鐭鍑芥暟f(x,y)鍦(0,0)瀵箈鐨勫亸瀵兼暟涓猴細lim(鈯縳->0) [f(0+鈯縳,0)-f(0,0)]/鈯縳 銆傚洜涓猴紝f(0,0) = 0 锛宖(鈯縳,0) = 锛堚娍x路0) / [(鈯縳)²+0²] = 0 锛涙墍浠ワ紝lim(鈯縳->0) [f(0+鈯縳,0)-f(0,0)]/鈯縳 = 0 锛涘嵆锛歠(x,y)鍦(...
绛旓細绉涓 f(x,y) 瀵 x (瀵 y )鐨勫亸瀵鍑芥暟銆傜畝绉板亸瀵兼暟銆傛寜鍋忓鏁扮殑瀹氫箟锛屽皢澶氬厓鍑芥暟鍏充簬涓涓嚜鍙橀噺姹傚亸瀵兼暟鏃讹紝灏卞皢鍏朵綑鐨勮嚜鍙橀噺鐪嬫垚甯告暟锛屾鏃朵粬鐨勬眰瀵兼柟娉曚笌涓鍏冨嚱鏁板鏁扮殑姹傛硶鏄竴鏍风殑銆傛瘮濡俧(x,y)=x^2+2xy+y^2锛瀵箈姹傚亸瀵灏辨槸f'x=(x^2)'+2y *(x)'=2x+2y銆
绛旓細瀵箈姹傚亸瀵灏辨槸f'x=(x^2)'+2y *(x)'=2x+2y 涓涓嚱鏁板湪鏌愪竴鐐圭殑瀵兼暟鎻忚堪浜嗚繖涓嚱鏁板湪杩欎竴鐐归檮杩戠殑鍙樺寲鐜囥傚鏁扮殑鏈川鏄氳繃鏋侀檺鐨勬蹇靛鍑芥暟杩涜灞閮ㄧ殑绾挎ч艰繎銆傚綋鍑芥暟f鐨勮嚜鍙橀噺鍦ㄤ竴鐐箈0涓婁骇鐢熶竴涓閲廻鏃讹紝鍑芥暟杈撳嚭鍊肩殑澧為噺涓庤嚜鍙橀噺澧為噺h鐨勬瘮鍊煎湪h瓒嬩簬0鏃剁殑鏋侀檺濡傛灉瀛樺湪锛屽嵆涓篺鍦▁0澶勭殑...
