Unless I'm making a mistake (which I probably am), when you differentiate (x \* |x|) / 2, you don't get |x|, which means (x \* |x|) / 2 isn't an antiderivative of |x|? 

Steps for differentiating (x |x|) / 2: 

First, note that the derivative of (|x| / 2) is x / (2 |x|). I got this just by using the fact that |x| = (x^2)^1/2 and applying the chain rule.

Split it up: (x \* |x|) / 2 = (x / 2) \* (|x| / 2) 

Apply the product rule: ( 1 \* (|x| / 2) ) + ( (x / 2) \* (x / 2 |x|) )

= (|x| \* 2) + (x^2 / 4|x|) 

So I'm getting the derivative as (|x| \* 2) + (x^2 / 4|x|) ... where's my mistake? I think I differentiated everything correctly. 

Any help is greatly appreciated.