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 , 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 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)