高数,大学高数,第68题,希望可以详细写出步骤,最好写在纸上,我很努力,

设y=tan(x+y),验证d³y/dx³=-2(3y^4+8y^2+5)/y^8
由y=tan(x+y),得
x+y=arctany,求导
1+y'=y'/(1+y²)
y'=(1+y²)/(-y²)
y''=2y'/y³
y'''=2(y''y³-y'*3y²*y')/y^6
=2(y''y-3y'²)/y^4
=2(2y'/y²-3y'²)/y^4
=2y'(2-3y'y²)/y^6
=2y'[2+3(1+y²)]/y^6
=2[(1+y²)/(-y²)](5+3y²)/y^6
=-2(1+y²)(5+3y²)/y^8
=-2(3y^4+8y^2+5)/y^8
即d³y/dx³=-2(3y^4+8y^2+5)/y^8,得证