Question

求隐函数的导数

x*e的y次方-y*e的x次方=y的x次方

Answer 1

左边对x求导
x'*e^y+x*(e^y)'-y'*e^x-y*(e^x)'
=e^y+x*e^y-y'*e^x-y*e^x

右边
令a=y^x
lna=xlny
对x求导
(1/a)*a'=lny+x*(1/y)*y'
a'=y^x*[lny+x*(1/y)*y']
所以e^y+x*e^y-y'*e^x-y*e^x=y^x*lny+y^x*(x/y)*y'
y^x*(x/y)*y'+y'*e^x=e^y+x*e^y-y*e^x-y^x*lny
所以y'=(e^y+x*e^y-y*e^x-y^x*lny)/[y^x*(x/y)+e^x]

Answer 2

先对方程两边却对数得
X*lne^y-Y*lne^x=lny^x
xy-yx=xlny
xlny=0开始求导
lny + x(1/y)y"=0
y"=-(ylny)/x 此题解完