FYI - Your professors comment was about what happens *WHILE* taking a derivative, not while algebraically simplifying a derivative you've already taken.
When you cancel common factors in fractions algebraically, you aren't doing anything different than cancelling common factors arithmetically.
As an aside, if you're ever unsure of some algebra rule, try thinking about a parallel arithmetic problem. It might prove enlightening. Remember that the algebra stuff only works because, hidden behind all those variables, it's all just numbers, so you're doing nothing different than arithmetic in disguise (well, until you get to calculus...).
You got to here (correctly it looked to me): .
You want to know if you can cancel *all* the (3-5x) bits in the numerator, giving this: ?
The answer is... no. For instance, plug in x=1 and you can see that those two are not equal.
In arithmetic, could you write . The answer is no, that's incorrect.
The way to simplify fractions using cancellation is to make sure that the factors you're cancelling multiply the ENTIRE rest of the numerator and denominator. Like this:
In your case:
And then simplify more: .