高等数学 多元函数微分学 求下列函数的偏导数 请给出详细步骤
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(1) u'<x> = y+1/y, u'<y> = x-x/y^2.
(2) u'<x> = [1√(x^2+y^2)-x*x/√(x^2+y^2)]/(x^2+y^2)
= (x^2+y^2-x^2)/(x^2+y^2)^(3/2) = y^2/(x^2+y^2)^(3/2).
u'<y> = -x*y/(x^2+y^2)^(3/2) = -xy/(x^2+y^2)^(3/2).
(3) u'<x> = (1/y)/[1+(x/y)^2] = y/(x^2+y^2).
u'<y> = (-x/y^2)/[1+(x/y)^2] = -x/(x^2+y^2).
(4) u'<x> = sin(x+y)+xcos(x+y), u'<y> = xcos(x+y).
(7) u = ln√(x^2+y^2) = (1/2)ln(x^2+y^2).
u'<x> = (1/2)2x/(x^2+y^2) = x/(x^2+y^2),
u'<y> = (1/2)2y/(x^2+y^2) = y/(x^2+y^2).
(2) u'<x> = [1√(x^2+y^2)-x*x/√(x^2+y^2)]/(x^2+y^2)
= (x^2+y^2-x^2)/(x^2+y^2)^(3/2) = y^2/(x^2+y^2)^(3/2).
u'<y> = -x*y/(x^2+y^2)^(3/2) = -xy/(x^2+y^2)^(3/2).
(3) u'<x> = (1/y)/[1+(x/y)^2] = y/(x^2+y^2).
u'<y> = (-x/y^2)/[1+(x/y)^2] = -x/(x^2+y^2).
(4) u'<x> = sin(x+y)+xcos(x+y), u'<y> = xcos(x+y).
(7) u = ln√(x^2+y^2) = (1/2)ln(x^2+y^2).
u'<x> = (1/2)2x/(x^2+y^2) = x/(x^2+y^2),
u'<y> = (1/2)2y/(x^2+y^2) = y/(x^2+y^2).
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