Question

求由方程x-y+ 1/2 siny=0所确定的隐函数y的二阶导数d^2y/dx^2

Answer 1

x-y+ 1/2 siny=0
F(x,y)=y-x-1/2siny=0
F,Fx,Fy在定义域的任意点都是连续的,
F(0,0)=0
Fy(x,y)>0
f'(x)=-Fx(x,y)/Fy(x,y)
=1/(1-1/2cosy)
=2/(2-cosy)
Fx(x,y)+Fy(x,y)y'=0
再求导:
Fxx(x,y)+Fxy(x,y)y'+[Fyx(x,y)+Fyy(x,y)y']y'+Fy(x,y)y''=0
所以
y''=[2FxFyFxy-F^2yFxx-F^2xFyy]/F^3y
将每一个偏导数分别求出来,再代入就可以了!
也可以对f'(x)对x求导
y'=f'(x)=2/(2-cosy)
这样比较容易一点
y''=[0+siny*y']/(2-cosy)^2
=2siny/(2-cosy)/(2-cosy)^2
=2siny/(2-cosy)^3
结果你检验一下