隐函数求二阶导的问题y^3+xy+x^2-2x+1=0 已知y(1)=0,y'(1)=0,求y''(1)的值.方程两边对x求导,得到3·y'·y^2+xy'+y+2x-2=0方程两边继续对x求导,得到6·y·y'^2+3·y''·y^2+y'+xy''+y'+2=0将x=1 y=0 y'=0 代入,解得y''(1)=-2
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![隐函数求二阶导的问题y^3+xy+x^2-2x+1=0 已知y(1)=0,y'(1)=0,求y''(1)的值.方程两边对x求导,得到3·y'·y^2+xy'+y+2x-2=0方程两边继续对x求导,得到6·y·y'^2+3·y''·y^2+y'+xy''+y'+2=0将x=1 y=0 y'=0 代入,解得y''(1)=-2](/uploads/image/z/10403271-63-1.jpg?t=%E9%9A%90%E5%87%BD%E6%95%B0%E6%B1%82%E4%BA%8C%E9%98%B6%E5%AF%BC%E7%9A%84%E9%97%AE%E9%A2%98y%5E3%2Bxy%2Bx%5E2-2x%2B1%3D0+%E5%B7%B2%E7%9F%A5y%281%29%3D0%2Cy%27%281%29%3D0%2C%E6%B1%82y%27%27%281%29%E7%9A%84%E5%80%BC.%E6%96%B9%E7%A8%8B%E4%B8%A4%E8%BE%B9%E5%AF%B9x%E6%B1%82%E5%AF%BC%2C%E5%BE%97%E5%88%B03%C2%B7y%27%C2%B7y%5E2%2Bxy%27%2By%2B2x-2%3D0%E6%96%B9%E7%A8%8B%E4%B8%A4%E8%BE%B9%E7%BB%A7%E7%BB%AD%E5%AF%B9x%E6%B1%82%E5%AF%BC%2C%E5%BE%97%E5%88%B06%C2%B7y%C2%B7y%27%5E2%2B3%C2%B7y%27%27%C2%B7y%5E2%2By%27%2Bxy%27%27%2By%27%2B2%3D0%E5%B0%86x%3D1+y%3D0+y%27%3D0+%E4%BB%A3%E5%85%A5%2C%E8%A7%A3%E5%BE%97y%27%27%281%29%3D-2)
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隐函数求二阶导的问题y^3+xy+x^2-2x+1=0 已知y(1)=0,y'(1)=0,求y''(1)的值.方程两边对x求导,得到3·y'·y^2+xy'+y+2x-2=0方程两边继续对x求导,得到6·y·y'^2+3·y''·y^2+y'+xy''+y'+2=0将x=1 y=0 y'=0 代入,解得y''(1)=-2
隐函数求二阶导的问题
y^3+xy+x^2-2x+1=0 已知y(1)=0,y'(1)=0,求y''(1)的值.
方程两边对x求导,得到3·y'·y^2+xy'+y+2x-2=0
方程两边继续对x求导,得到6·y·y'^2+3·y''·y^2+y'+xy''+y'+2=0
将x=1 y=0 y'=0 代入,解得y''(1)=-2 但正确答案为-1/2.若先方程两边对x求导,然后求出y'(x,y),再单独对y'(x,y)的表达式求导,得出的结果就是-1/2.
那么这种方法错在哪里呢?
隐函数求二阶导的问题y^3+xy+x^2-2x+1=0 已知y(1)=0,y'(1)=0,求y''(1)的值.方程两边对x求导,得到3·y'·y^2+xy'+y+2x-2=0方程两边继续对x求导,得到6·y·y'^2+3·y''·y^2+y'+xy''+y'+2=0将x=1 y=0 y'=0 代入,解得y''(1)=-2