Not necessarily true. A good rule for introductory math students, but some advanced math requires dividing by zero. (As mentioned, that's what a derivative is, a division by zero.)
Limits are a way of getting information out of a division by zero, which is why derivatives involve taking the limit.
Division by zero is kind of like the square root of a negative number (something introductory mathematics coursework also tells you not to do). It's not an invalid operation, it's just an operation you have to be aware of the ramifications of. (If it seems like zero has unusual behavior, well, the same is true of negative numbers with respect to zero and positive numbers, and again the same is true of positive numbers with respect to zero and negative numbers.)
Another link, this time explicitly dealing with derivatives and division by zero, in the vain hope that you'll actually update someday.
If it's worth saying, but not worth its own post (even in Discussion), then it goes here.