设z=xf(x+y)+g(xy,x2+y2),其中f,g分别二阶连续可导和二阶连续可偏导,则________.

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问题 设z=xf(x+y)+g(xy,x2+y2),其中f,g分别二阶连续可导和二阶连续可偏导,则________.

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答案f′+xf″+xy-1g′1+yxy-1lnxg′1+yx2y-1lnxg″11+2y2xy-1g″12+2xy+1lnxg″21+4xyg″22

解析 由z=xf(x+y)+g(xy,x2+y2),得
=f(x+y)+xf′(x+y)+yxy-1g′1(xy,x2+y2)+2xg′2(xy,x2+y2),
=f′+xf″+xy-1g′1+yxy-1lnxg′1+yx2y-1lnxg″11+2y2xy-1g″12+2xy+1lnxg″21+4xyg″22.
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