高数 求y=tan(x+y) 的二阶导数？

y＝tan(x+y)两边求导：y'＝(sec(x+y))^2×(1+y')，所以y'＝－(csc(x+y))^2＝－1－1/y^2

所以，y''＝2y'/y^3＝－2(1+y^2)/y^5

y=tan(x+y)

y'=tan'(x+y)
=sec^2(x+y)(x+y)'
=sec^2(x+y)*(1+y')

y'=sec^2(x+y)/[1-sec^2(x+y))
=-sec^2(x+y)/tan^2(x+y)
=-1/sin^2(x+y)
=-csc^2(x+y)

y''=-2csc(x+y)*[csc(x+y)]'
=-2csc(x+y)*[-csc(x+y)cot(x+y)](x+y)'
=2csc^2(x+y)cot(x+y)(1+y')
=2csc^2(x+y)cot(x+y)[1-csc^2(x+y)].