Derivative

**ilovemymath** · August 8th 2010, 07:50 PM

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

**General** · August 8th 2010, 08:12 PM

Do you know the limit definition of the derivative of the function at point ?

**ilovemymath** · August 8th 2010, 08:18 PM

nop

**eumyang** · August 8th 2010, 08:42 PM

Hmm... you should! The derivative of f at c is
$f'(c) = \lim\limits_{x \to c} \dfrac{f(x) - f(c)}{x - c}$ .

The existence of this limit in this form requires that the one-sided limits
$f'(c) = \lim\limits_{x \to c-} \dfrac{f(x) - f(c)}{x - c}$ and $f'(c) = \lim\limits_{x \to c+} \dfrac{f(x) - f(c)}{x - c}$ 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.

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