I'm asked to find the derivative of f(x)=x|x| at the point (0,0).
Would I do this using the definition of absolute value? o_O
I would start this problem by piecewise defining $\displaystyle f(x)=(x)|x|$, which I think is what you meant by the definition of the absolute value. Then take the derivatives of each piecewise defined function. Looking at the graph of $\displaystyle f(x)=(x)|x|$ will reveal that the function has no "problems" at x=0, i.e. it is continuous and smooth in the neigborhood surrounding the point x=0.
Hope this helps.
(P.S. I'm not sure if this is correct but this is where I would start if I were doing the problem)
Do you see that $\displaystyle \begin{array}{lcl}
{x > 0} & \Rightarrow & {x\left| x \right| = x^2 } \\
{x < 0} & \Rightarrow & {x\left| x \right| = - x^2 } \\
\end{array} $?
You should take a good look at the graph of $\displaystyle {x\left| x \right|}$.