Question

求隐函数的二阶导数，第三题和第四题

求隐函数的二阶导数，第三题和第四题求大神解答一下，如图

Answer 1

y=tan(x+y)
y'=sec^2(x+y)(1+y')
y'(1-sec^2(x+y)=sec^2(x+y)
y'=-sec^2(x+y)/(sec^2(x+y-1)
=-sec^2(x+y)/tan^2(x+y)
=-csc^2(x+y)
y''=-2csc(x+y)*[-csc(x+y)*ctg(x+y)]*(1+y')
=2csc^2(x+y)ctg(x+y)[1-csc^2(x+y)]
=-2csc^2(x+y)ctg^3(x+y)

y=1+xe^y
y'=e^y+xe^yy'
y'(1-xe^y)=e^y
y'=e^y/(1-xe^y)
y''=[e^yy'(1-xe^y)+e^y(e^y+xe^yy')]/(1-xe^y)^2
=e^y[y'(1-xe^y)+(e^y+xe^yy')]/(1-xe^y)^2
=e^y[e^y/(1-xe^y)*(1-xe^y)+(e^y+xe^y*e^y/(1-xe^y))]/(1-xe^y)^2
=e^y[e^y+(e^y+xe^y*e^y/(1-xe^y))]/(1-xe^y)^2
=e^2y[1+(1+xe^y/(1-xe^y)]/(1-xe^y)^2
=e^2y[(1-xe^y)+(1+xe^y)]/(1-xe^y)^3
=2e^2y/(1-xe^y)^3