y=tan(x-y) 求二阶导数

Question

高数 求 y=tan(x-y) 二阶导数的详细过程，谢谢！！！

Answer 1

y=tan(x-y)两边求导
y' = (1-y')/cos²(x-y)
y' = 1/[1+cos²(x-y)]
再次求导
y" = -2cos(x-y)[-sin(x-y)](1-y')/[1+cos²(x-y)]²
= sin(2x-2y)(1-y')/[1+cos²(x-y)]²
= sin(2x-2y){1 - 1/[1+cos²(x-y)]}/[1+cos²(x-y)]²
= sin(2x-2y)cos²(x-y)/[1+cos²(x-y)]³

Answer 2

由y=tan（x-y）
y′=1/cos²（x-y）×（1-y′）
=1/cos²（x-y)-y′/cos²(x-y)
y′+y′/cos²(x-y)=1/cos(x-y)
∴y′=1/[cos²(x-y)+1]
y″=-2cos(x-y)(1-y′)/[cos²(x-y)+1]².

Answer 3

y' = (1-y')/cos²(x-y)
y' = 1/[1+cos²(x-y)]
y" = -2cos(x-y)[-sin(x-y)](1-y')/[1+cos²(x-y)]²
= sin(2x-2y)(1-y')/[1+cos²(x-y)]²
= sin(2x-2y){1 - 1/[1+cos²(x-y)]}/[1+cos²(x-y)]²
= sin(2x-2y)cos²(x-y)/[1+cos²(x-y)]³

Answer 4

sin（x-y）/cos（x-y）