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Partial Differentiation 3
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To:laisai yu ?u/?x=2x, 但 ?x/?u不是1/(2x) (x+y)^2=u+2xy=(vxy)^2, 則(vxy- 1/v)^2=u+1/v^2 vxy= 1/v +/- √(u+ 1/v^2) (負不合,因x,y>0) 故xy= [1+√(uv^2+1)]/v^2 x+y=[1+√(uv^2+1)]/v (x-y)^2=u-4xy=u- 4√[1+(uv^2+1)]/v^2 (i) lnz=w+x+y (1/z)(?z/?x)=(?w/?u)*(?u/?x)+(?w/?v)*(?v/?x)+1 =(?w/?u)*(2x)+(?w/?v)*(-1/x^2)+1 ----(1) (1/z)(?z/?y)=(?w/?u)*(?u/?y)+(?w/?v)*(?v/?y)+1 =(?w/?u)*(2y)+(?w/?v)*(-1/y^2)+1 ----(2) (ii) y*(1)-x*(2)得 [y(?z/?x)-x(?z/?y)]/z=(?w/?u)*0-(?w/?v)*(y/x^2-x/y^2)+y-x (y-z)=(?w/?v)*(x^3-y^3)/(x^2y^2)+y-x 則 (?w/?v)(x^3-y^3)=0 又x, y不相等, 故?w/?v=0
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Partial Differentiation 3
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最佳解答:To:laisai yu ?u/?x=2x, 但 ?x/?u不是1/(2x) (x+y)^2=u+2xy=(vxy)^2, 則(vxy- 1/v)^2=u+1/v^2 vxy= 1/v +/- √(u+ 1/v^2) (負不合,因x,y>0) 故xy= [1+√(uv^2+1)]/v^2 x+y=[1+√(uv^2+1)]/v (x-y)^2=u-4xy=u- 4√[1+(uv^2+1)]/v^2 (i) lnz=w+x+y (1/z)(?z/?x)=(?w/?u)*(?u/?x)+(?w/?v)*(?v/?x)+1 =(?w/?u)*(2x)+(?w/?v)*(-1/x^2)+1 ----(1) (1/z)(?z/?y)=(?w/?u)*(?u/?y)+(?w/?v)*(?v/?y)+1 =(?w/?u)*(2y)+(?w/?v)*(-1/y^2)+1 ----(2) (ii) y*(1)-x*(2)得 [y(?z/?x)-x(?z/?y)]/z=(?w/?u)*0-(?w/?v)*(y/x^2-x/y^2)+y-x (y-z)=(?w/?v)*(x^3-y^3)/(x^2y^2)+y-x 則 (?w/?v)(x^3-y^3)=0 又x, y不相等, 故?w/?v=0
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