三类含绝对值的函数的可导性

Differentiability of three types of functions with absolute value

使用导数定义以及数学归纳原理,探讨了三类含绝对值的函数的可导性,证明了(1)若y=|f(x)|在x_(0)点处可导,则y=f(x)在x_(0)点的可导性取决于f(x_(0))与f’(x_(0));(2)对于任意的正整数k,y=(x-a)^(k)|x-a|在x=a处具有k阶导数,不具有k+1阶导数;(3)若g(x)在x=a处连续,则y=|x-a|g(x)在x=a处的可导性取决于g(a).

By using the definition of derivative and the principle of mathematical induction,the differentiability of three kinds of functions with absolute values is discussed.It is proven that(1)If y=f(x)is derivable at the point x_(0),then the differentiability of y=f(x)at poin x_(0)depends on f(x_(0))and f'(x_(0));(2)For any positive integer k y=(x-a)^(k)|x-a|has a k order derivative at the point a,and no k+1 order derivative;(3)If g(x)is continuous at the point a,then the differentiability of y=|x-a|g(x)at point a depends on g(a).