$$ f(x) = u(x)/v(x) $$
$$ f'(x)= $$
$$ (u'(x)*v(x)-u(x)*v'(x))/v(x)^2 $$
$$ u(x)= (x-1)^{\frac{1}{3}} $$
$$ u'(x)=1/3* (x-1)^{\frac{-2}{3}} $$
$$ v(x)= (x+3)^{\frac{1}{3}} $$
$$ v'(x)=1/3* (x+3)^{\frac{-2}{3}} $$
$$ f'(x)= $$
$$ (1/3* (x-1)^{\frac{-2}{3}} * (x+3)^{\frac{1}{3}} $$
$$ - (x-1)^{\frac{1}{3}} * 1/3* (x+3)^{\frac{-2}{3}} ) / $$
$$ (x+3)^{\frac{2}{3}} $$
$$ = 1/3*( (x-1)^{\frac{-2}{3}} * (x+3)^{\frac{-1}{3}} $$
$$ - (x-1)^{\frac{1}{3}} * (x+3)^{\frac{--4}{3}} ) $$
Gut möglich, dass es noch einfacher geht.
:-)