已知隐函数XY=e(X+Y)次方,求dy

已知隐函数XY=e(X+Y)次方,求dy
已知隐函数XY=e(X+Y)次方,求dy

已知隐函数XY=e(X+Y)次方,求dy
一：∵xy=e^(x+y) ==>d(xy)=d(e^(x+y)) (两端取微分)
==>xdy+ydx=e^(x+y)(dx+dy)
==>xdy+ydx=e^(x+y)dx+e^(x+y)dy
==>xdy-e^(x+y)dy=e^(x+y)dx-ydx
==>(x-e^(x+y))dy=(e^(x+y)-y)dx
∴dy=[(e^(x+y)-y)/(x-e^(x+y))]dx；

先将等式两边同时取对数：ln（xy）=x+y
两边同时求导：（y+xy')/xy=1+y'
整理得y'=（y-xy）/(xy-1)
因为y'=dy/dx
所以dy=[（y-xy）/(xy-1)]dx

xy=e^(x+y),
∵xy=e^(x+y)
∴d(xy)=d[e^(x+y)]
∴y+xdy/dx=d(x+y)e^(x+y)=(1+dy/dx)e^(x+y)
∴(x-e^(x+y))dy/dx=e^(x+y)-y
∴dy/dx=[e^(x+y)-y] / [x-e^(x+y)]
dy=[e^(x+y)-y] / [x-e^(x+y)]dx

x y = e^(x+y)
求导：y + x * y' = e^(x+y) * (1 + y')
即： y + x * y' = x y * (1 + y')
解得： y' = (xy - y) / (x - xy)
dy = [(xy - y) / (x - xy)] * dx

XY=e(X+Y) Y=e(X+Y)/X
dy=de(X+Y)/X
dy=de+dey/x

函数XY=e^(X+Y)
对x求导
y+xy'=e^(X+Y)（1+y‘）
y+xy'=e^(X+Y)+y‘e^(X+Y)
xy'-y‘e^(X+Y)=e^(X+Y)-y
y'【x--e^(X+Y)】=e^(X+Y)1-y
y' =【e^(X+Y)-y】/【x--e^(X+Y)】
所以 dy={【e^(X+Y)-y】/【x--e^(X+Y)】}dx
希望对你有所帮助，祝你学习进步！

原式变为：ln(xy)＝ln（e的x+y次方）
lnx+lny=x+y
两边对x求导得：1/x+dy/ydx=1+dy/dx
解得：dy/dx=y(x-1)/x(1-y)

见图，将dx移过去即可

e^y-xy=e
e^y·dy/dx-(y+x·dy/dx)=0
e^y·dy/dx-y-x·dy/dx=0
(e^y-x)·dy/dx=y
dy/dx=y/(e^y-x)
dy/dx不能叫做dx分之dy,因为这是个导数符号,而不是分数!

解法一：∵xy=e^(x+y) ==>d(xy)=d(e^(x+y)) (两端取微分)
==>xdy+ydx=e^(x+y)(dx+dy)
==>xdy+ydx=e^(x+y)dx+e^(x+y)dy
...

全部展开

解法一：∵xy=e^(x+y) ==>d(xy)=d(e^(x+y)) (两端取微分)
==>xdy+ydx=e^(x+y)(dx+dy)
==>xdy+ydx=e^(x+y)dx+e^(x+y)dy
==>xdy-e^(x+y)dy=e^(x+y)dx-ydx
==>(x-e^(x+y))dy=(e^(x+y)-y)dx
∴dy=[(e^(x+y)-y)/(x-e^(x+y))]dx；
解法二：∵xy=e^(x+y) ==>xy=e^x*e^y
==>ye^(-y)=e^x/x
==>d(ye^(-y))=d(e^x/x) (两端取微分)
==>(e^(-y)-ye^(-y))dy=((xe^x-e^x)/x²)dx
==>dy=[(xe^x-e^x)/(x²(e^(-y)-ye^(-y)))]dx
==>dy=[(xe^(x+y)-e^(x+y))/(x(x-xy))]dx (分子分母同乘e^y)
==>dy=[(xe^(x+y)-xy)/(x(x-e^(x+y)))]dx (∵e^(x+y)=xy)
==>dy=[x(e^(x+y)-y)/(x(x-e^(x+y)))]dx
∴dy=[e^(x+y)-y)/(x-e^(x+y))]dx。

收起

曾经的我也被这东西深深的折磨，对高数简直深恶痛绝，但是现在哥把它全都就饭吃了。

这种题很简单啊！！
前提是不要紧张
函数两边对x求导数就可以了e^(xy)=x+y+e-2；等式两边对x求导
得左边为d（e^(xy)）=e^(xy)*y*dx+e^(xy)*x*dy
右边=dx+dy，则有e^(xy)*y*dx+e^(xy)*x*dy=dx+dy整理即可解出dy/dx；

ln（xy）=x+y
（y+xy')/xy=1+y'
y'=（y-xy）/(xy-1)
y'=dy/dx
dy=[（y-xy）/(xy-1)]dx

xy=e^(x+y)
ln(xy)=ln(e^(x+y))
d(lnx+lny)=d(x+y)
dx/x+dy/y=dx+dy
(1/y-1)dy=(1-1/x)dx
dy=(1-1/x)/(1/y-1)dx
dy=(xy-y)/(x-xy)dx

XY=e(X+Y)次方
dXY=de(X+Y)次方
xdy+ydx=e(X+Y)次方*d(x+y)
xdy+ydx=e(X+Y)次方*dx+e(X+Y)次方*dy
dy(x-e(X+Y)次方)=(e(X+Y)次方-y)dx
dy=)=[(e(X+Y)次方-y)dx]/(x-e(X+Y)次方）
微分不变性

XY=e^(X+Y)
Y+Xdy=(1+dy)*e^(X+Y)
dy=(y-e^(X+Y))/[e^(X+Y)-x]