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?

You can derive the quotient rule by considering $\dfrac{f(x)}{g(x)}$ as $f(x)\cdot(g(x))^{-1}$ and then using the product and chain rule.The quotient rule  gives a formula (under the right conditions) for evaluating  the derivative of  a quotient without using the product and chain rule each time.
