$h\left(x\right)=\frac{f\left(x\right)}{g\left(x\right)}$  

$h\left(x\right)=\frac{x^2-6}{x^2+6}$  

$h{}^{\prime }\left(x\right)=\frac{\left(x^2-6\right){}^{\prime }\cdot \left(x^2+6\right)-\left(x^2-6\right)\cdot \left(x^2+6\right){}^{\prime }}{{\left(x^2+6\right)}^2}$