题目

求y=tan（x+y）的二阶导数

提问时间：2020-09-03

答案

y=tan(x+y)
y'=sec²(x+y)*(x+y)'
=sec²(x+y)*(1+y')
=sec²(x+y)+y'sec²(x+y)
y'-y'sec²(x+y)=sec²(x+y)
y'=sec²(x+y)/[1-sec²(x+y)]
=sec²(x+y)/{-[sec²(x+y)-1]}
=sec²(x+y)/[-tan²(x+y)]
=-1/cos²(x+y)*cos²(x+y)/sin²(x+y)
=-csc²(x+y)
y''=-2csc(x+y)*[-csc(x+y)cot(x+y)]*(x+y)'
=2csc²(x+y)cot(x+y)*(1+y')
=2csc²(x+y)cot(x+y)*[1-csc²(x+y)]
=2csc²(x+y)cot(x+y)*{-1[csc²(x+y)-1]}
=-2csc²(x+y)cot(x+y)*[cot²(x+y)]
=-2csc²(x+y)cot³(x+y)