Answer
$8x-\displaystyle \frac{12|x|}{x}$
Work Step by Step
SUMMARY (rules in differential notation):
1. The Power Rule$:\ \ \ \displaystyle \frac{d}{dx}[x^{n}]=n\cdot x^{n-1 } $
2. Sum Rule: $\displaystyle \ \ \ \frac{d}{dx}[f\pm g](x)=\frac{d}{dx}[f(x)]\pm\frac{d}{dx}[g(x)] $
3. Constant Multiple Rule:$\ \ \displaystyle \frac{d}{dx}[cf(x)]=c\cdot\frac{d}{dx}[f(x)] $
4. Constant times x:$\ \ \ \displaystyle \frac{d}{dx}(cx)=c $
5. Constant:$\displaystyle \ \ \ \ \ \frac{d}{dx}(c)=0 $
6. $\displaystyle \frac{d}{dx}(|\mathrm{x}|)=\frac{|x|}{x}$, (from sec.10.6)
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$ \displaystyle \frac{d}{dx}[4(x^{2}+3|x|)$ = $\ \ \ $...(2)
$=\displaystyle \frac{d}{dx}(4x^{2})-\frac{d}{dx}(12|x|)$ = $\ \ \ $...($3$)
$=4\displaystyle \frac{d}{dx}( x^{2})-12\frac{d}{dx}(|x|)$ = $\ \ \ $...($1,\ \ 6 $)
$=4(2x)-12\displaystyle \frac{|x|}{x}$
$=8x-\displaystyle \frac{12|x|}{x}$
