# Derivative

• Aug 8th 2010, 08:50 PM
ilovemymath
Derivative
For the function $f (x)=x|x|$ , show that $f ' (0)$ exists. What is the value?
• Aug 8th 2010, 09:12 PM
General
Do you know the limit definition of the derivative of the function at point ?
• Aug 8th 2010, 09:18 PM
ilovemymath
nop
• Aug 8th 2010, 09:42 PM
eumyang
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.