Assume that prime (') means d/dx.
g(6)=3; g'(6)=-4
d(u/v)/dx=(vdu/dx-udv/dx)/v^2. Let u=x^4 then du/dx=4x^3; let v=g(x) then dv/dx=g'(x).
p(x)=x^4/g(x)=u/v; p'(x)=(4x^3g(x)-x^4g'(x))/(g(x))^2.
For x=6: p'(6)=(4*216g(6)-1296g'(6))/(g(6))^2=(864*3-1296(-4))/9=(2592+5184)/9=7776/9=864.