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 x0,then the differentiability of v=|f(x)| at poinx0 depends on f(x0)and f'(x0);(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).
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