"The carrot juice."
Translation:Ο χυμός καρότο.
When you have compound nouns like this, inserting an imaginary από (from) also helps.
So, you would have χυμός από καρότο - you wouldn't have the opposite construction, which would be 'carrot from juice'.
Whatever sounds correct using με/από usually is, and also determines which article to use if the nouns have different genders, as in the case above. The 'root' word which you place first determines the gender.
Another helpful user explained this to me a week ago - I was a bit confused too ;-)
They are both correct, and one could argue that, between the two, χυμός καρότου is the most grammatically correct. It's just not as used in everyday life. You'll hear more often "(θέλω) ένα χυμό πορτοκάλι", for example (rather than "πορτοκαλιού"), but it's also correct. The most correct, in fact.