When we are forced to derive something like:

dy   9x2
dx   3x

We could divide this out or re-write it to allow us to use the product rule, but the simplest way is to use what is called the quotient rule:

(f/g)'(x) = g(x)f'(x)-f(x)g'(x)
All this means is that we {(multiply the denominator with the derivative of the numerator), subtract the product of (the numerator and the derivative of the denominator)} and divide all this mess by the {square of the denominator}.
There is a very silly, but handy, mnemonic for remembering this formula. It runs as follows:

If you pretend you're Santa Claus, you could call f(x) "Hi" (as in, high, on top), and g(x) "Ho" (rhymes with "low"). Then the formula is "HoDeeHi minus HiDeeHo over HoHo".

(dy / dx) = HoDeeHi - HiDeeHo
You will probably feel like an idiot reciting this to calculus students, but hey, whatever works.

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