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)
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绛旓細濡傚浘鎵绀猴細
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