Question

高数隐函数二阶求导

高数隐函数二阶求导第四题 二阶导数为什么我和答案算得不一样





Answer 1

y=1+xe^y
y'= e^y + (xe^y)y'
(1-xe^y)y' = e^y
y' =e^y/(1-xe^y)
y''

=[e^y/(1-xe^y)]y' - [ e^y/(1-xe^y)^2] .[ -e^y - xe^y.y']
=[e^y/(1-xe^y)].[e^y/(1-xe^y)] - [ e^y/(1-xe^y)^2] .{ -e^y - xe^y.[e^y/(1-xe^y)] }
=e^(2y)/(1-xe^y)^2 - [ e^y/(1-xe^y)^2] .[ -e^y /(1-xe^y)]

=e^(2y)/(1-xe^y)^2 + e^(2y)/(1-xe^y)^3

Answer 2

两边求导
y'=e^y+y'xe^y
y'(1-xe^y)=e^y
y''(1-xe^y)-y'(e^y+y'xe^y)=y'e^y
y''(1-xe^y)-y'y'=y'e^y
y''(1-xe^y)-[e^y/(1-xe^y)]^2=e^2y/(1-xe^y)
y''=e^2y/(1-xe^y)^3+e^2y/(1-xe^y)^2