Why is the quotient rule in differentiation necessary? 
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*Calculus - Derivatives - Quotient Rule
Why is a quotient rule even necessary?
Why can't we just consider $\frac{A}{B}$ as $A \cdot B^{-1}$ and use the multiplication formula?

Personally, I have never used the quotient rule as I have always found it ugly and inconvenient. For example try computing the derivative of $\large\frac{(x+1)^3}{(2x-4)^5}$ with respect to $x$. Since the quotient rule can be derived from the product rule and chain rule, it is indeed redundant if you have the other two as well as the derivative of $x^{-1}$ with respect to $x$.
One might point out that we could just as well say that all the rules are redundant if you have the definition of derivative. However, there is something called the formal derivative that is defined algebraically and not via a limit-based definition (because it makes no sense in that context). So there is some benefit (beyond just for ease of computation) in finding purely algebraic rules for differentiation.
