$f\left(g\left(x\right)\right)=\left\{\begin{array}{ll}g\left(x\right)+2,& g\left(x\right)<0\\ {\left(g\left(x\right)\right)}^2,& g\left(x\right)\ge 0\end{array}\right.$

$=\left\{\begin{array}{ll}{x}^3+2,& x<0\\ x^6,& x\in [0,1)\\ {\left(3x-2\right)}^2,& x\in [1,\infty )\end{array}\right.$

${\left(fog\left(x\right)\right)}^{\text{'}}=\left\{\begin{array}{ll}3x^2,& x<0\\ 6x^5,& x\in [0,1)\\ 2\left(3x-2\right)\times 3,& x\in [1,\infty )\end{array}\right.$

At $x=0$

$\mathrm{L}.\mathrm{H}.\mathrm{L}.\ne \mathrm{R}.\mathrm{H}.\mathrm{L}.$  (Discontinuous)

At $x=1$

$\mathrm{L}.\mathrm{H}.\mathrm{L}.=\mathrm{R}.\mathrm{H}.\mathrm{L}.$  (continuous)

$\mathrm{L}.\mathrm{H}.\mathrm{D}.=6=\mathrm{R}.\mathrm{H}.\mathrm{D}.$

$\Rightarrow fog\left(x\right)$ is differentiable for all $x\in R-\left\{0\right\}$