一道关于反函数导数的问题
来源:学生作业帮 编辑:搜搜做题作业网作业帮 分类:数学作业 时间:2024/07/17 13:42:01
一道关于反函数导数的问题
若f-1(x)是f(x)的反函数,G(x)=1/f-1(x).f(3)=2,f'(3)=1/9 .求G'(2)=?
若f-1(x)是f(x)的反函数,G(x)=1/f-1(x).f(3)=2,f'(3)=1/9 .求G'(2)=?
![一道关于反函数导数的问题](/uploads/image/z/19862500-4-0.jpg?t=%E4%B8%80%E9%81%93%E5%85%B3%E4%BA%8E%E5%8F%8D%E5%87%BD%E6%95%B0%E5%AF%BC%E6%95%B0%E7%9A%84%E9%97%AE%E9%A2%98)
dy/dx=f'(x)
dx/dy=1/f'(x)
[f-1(x)]'=1/f'(y) (x,y互换过了)
G(x)=1/f-1(x)
[G(x)]'=-1/[f-1(x)]^2*[f-1(x)]'=-1/[f-1(x)]^2*1/f'(y)
G'(2)=-1/[f-1(2)]^2*1/f'(y)
=-1/(3^2)*1/(1/9)
=-1.
dx/dy=1/f'(x)
[f-1(x)]'=1/f'(y) (x,y互换过了)
G(x)=1/f-1(x)
[G(x)]'=-1/[f-1(x)]^2*[f-1(x)]'=-1/[f-1(x)]^2*1/f'(y)
G'(2)=-1/[f-1(2)]^2*1/f'(y)
=-1/(3^2)*1/(1/9)
=-1.