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_Am I taking the antiderivative of |x| correctly?_ That depends on how rigorous you want to be. From a technical point of view, no.

_How would I take it?_  The main problem of your argumentation is that you derivate $\left|\cdot\right|$ which isn't differentiable ([S06], pages 142 and 143) . Instead, you can do the following. Let $f:=\left|\cdot\right|$ and $id$ be the identity function: if $x\ge0, F_0(x)=\int _0^xf=\int _0^xid=x^2/2$; if $x<0, F_0(x)=\int _0^xf=\int _0^x-id=-x^2/2$. This function is an antiderivative of the absolute value function by the First Fundamental Theorem of Calculus ([S06], page 268). If you want to condense its formula, you can write $F_0(x)=(x\left|x\right|)/2$, for every $x\in R$. To obtain the indefinite (which seems to be what you want) antiderivative, you can write $F(x)=(x\left|x\right|)/2+c$, where $c\in R$ (which is actually a notation for $F=\{F_c:c\in R\}$ where $F_c(x)=(x\left|x\right|)/2+c$).

[S06] Michael D. Spivak, _Calculus_, 3rd edition (2006).