Question

隐函数,dy/dx=(x+y)/(x-y),求y关于x的二阶导.答案是2（x^2+y^2）/(x-y)^3.

Answer 1

y'=(x+y)/(x-y)

xy'-yy'=x+y

x+yy'=xy'-y

两边求导

1+yy''+y'y'=y'+xy''-y'

1+yy''+y'y'=xy''

1+y'y'=(x-y)y''

y''=(1+y'y')/(x-y)

将y'=(x+y)/(x-y)代入

得

y''=(1+(x+y)^2/(x-y)^2)/(x-y)

=((x-y)^2+(x+y)^2)/(x-y)^3

=2(x^2+y^2)/(x-y)^3