For the function $\displaystyle f (x)=x|x|$ , show that $\displaystyle f ' (0)$ exists. What is the value?

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- Aug 8th 2010, 07:50 PMilovemymathDerivative
For the function $\displaystyle f (x)=x|x|$ , show that $\displaystyle f ' (0)$ exists. What is the value?

- Aug 8th 2010, 08:12 PMGeneral
Do you know the limit definition of the derivative of the function at point ?

- Aug 8th 2010, 08:18 PMilovemymath
nop

- Aug 8th 2010, 08:42 PMeumyang
Hmm... you should! The derivative of f at c is

$\displaystyle f'(c) = \lim\limits_{x \to c} \dfrac{f(x) - f(c)}{x - c}$.

The existence of this limit in this form requires that the one-sided limits

$\displaystyle f'(c) = \lim\limits_{x \to c-} \dfrac{f(x) - f(c)}{x - c}$ and $\displaystyle f'(c) = \lim\limits_{x \to c+} \dfrac{f(x) - f(c)}{x - c}$ exist and are equal. I think you will have to evaluate both one-sided limits and show that they are equal in order to prove that f'(0) exists in your case.