Question

求由下列方程所确定的隐函数的导数。

1.3.5.7.9题

Answer 1

1. 方程xy=e^(x+y)确定的隐函数y的导数：

y'=[e^(x+y)-y]/[x-e^(x+y)]

解题过程：

方程两边求导：　

y+xy'=e^(x+y)(1+y')

y+xy'=e^(x+y)+y'e^(x+y)　

y'[x-e^(x+y)]=e^(x+y)-y　

得出最终结果为：

y'=[e^(x+y)-y]/[x-e^(x+y)]

追答

5. 两边对x求导
则1*cosy+x*(-siny)*y'=cos(x+y)*(1+y')
cosy-xsiny*y'=cos(x+y)+cos(x+y)*y'
所以y'=[cosy-cos(x+y)]/[xsiny+cos(x+y)]

7. e^y=cos(x+y)
(e^y).y' = -sin(x+y) .( 1+ y')
(e^y + sin(x+y) ) y' = -sin(x+y)
y' =-sin(x+y)/(e^y + sin(x+y) )

9. -e^x+e^y-xy=0
两边同时对x求导,得-e^x+y'e^y-y-xy'=0 y'=(y+e^x)/(-x+e^y)

3.y^2+3xy+8=0

2yy'+3y+3xy'=0
y'=-3y/3x+2y