若由方程y=tan(x+y)所确定的隐函数为y=y(x),求y''(x)是二阶导数y''(x),

Question

若由方程y=tan(x+y)所确定的隐函数为y=y(x),求y''(x)
是二阶导数y''(x),

Answer 1

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)