The function in the general form:
#y=ln(f(x))#
then its derivative is
#y'=1/f(x)*f'(x)#
where #f(x)=(color(red)g(x))^5#
and its derivative is:
#f'(x)=5(color(red)g(x))^4*g'(x)#
where #color(red)(g(x)=-(4x^4)/(x^3-3))#
and its derivative is:
#g'(x)=((-4*4x^3)(x^3-3)-(-4x^4)(3x^2))/(x^3-3)^2#
#=(-16x^6+48x^3+12x^6)/(x^3-3)^2#
#=(48x^3-4x^6)/(x^3-3)^2=(4x^3(12-x^3))/(x^3-3)^2#
Finally, it is:
#y'=1/(color(red)g(x))^5*5(color(red)g(x))^4*g'(x)#
#=1/(-(4x^4)/(x^3-3))^cancel5*5cancel((-(4x^4)/(x^3-3))^4)*(4x^3(12-x^3))/(x^3-3)^2#
#=-(5cancel((x^3-3)))/(cancel4x^cancel4)*(cancel(4x^3)(12-x^3))/(x^3-3)^cancel2#
#=-(5(12-x^3))/(x(x^3-3))#
