The answer to
> How does a Physicist and Mathematician solve this type question? Even,
> is it OK to use Product rule instead of Quotient rule in University
> and Real Life?
is that any experienced scientist knows several methods to solve problems and uses those that are most convenient for them at that particular time.
I would look at that derivative and use the quotient rule. But if there was something in the source of the problem that suggested that it made more sense to write the denominator as $(3x^2+2x^3)^{-1}$ then the product rule would be more appropriate.
