求z=(y/x)^x/y(y/x的y/x次方)在(1,2)对x的偏导数

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求z=(y/x)^x/y(y/x的y/x次方)在(1,2)对x的偏导数
求z=(y/x)^x/y(y/x的y/x次方)在(1,2)对x的偏导数

求z=(y/x)^x/y(y/x的y/x次方)在(1,2)对x的偏导数
z=(y/x)^(x/y) 是y/x的x/y次方吧
两边取对数
lnz=(x/y)ln(y/x)
lnz=(1/y)*[xln(y/x)],两边对x求导,把y当常数
az/ax*1/z=(1/y)*[ln(y/x)+x*(y*-1/x²)/(y/x)]
az/ax=(1/y)*[ln(y/x)-(x*y/x²)*(x/y)]*z
az/ax=(1/y)*[ln(y/x)-1]*(y/x)^(x/y)
∴az/ax|(x=1,y=2)
=(1/2)*[ln(2/1)-1]*(2/1)^(1/2)
=(1/2)*(ln2-1)*√2
=(ln2-1)*√2/2*(√2/√2)
=(ln2-1)/√2,(根号2)分之(ln2-1)

z=(y/x)^x/y=e^ln[(y/x)^(x/y)]=e^[x/y*ln(y/x)]=e^[x/y*(lny-lnx) ]
dz/dx=e^[x/y*(lny-lnx)] *[1/y*(lny-lnx)+(y/x)(-1/x)]

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