Convention verses memory: The quotient rule v product rule for derivatives I have long wondered why the product rule is taught the way it is. ${ d(UV)=Udv+Vdu}$ 
Don't get me wrong, I am not a complete NOB when it comes to calc, but the quotient rule states $${d(\frac {U}{V})=\frac {Vdu-Udv}{V^2}}$$ I know this is a matter of semantics, but is just seems to me that (in order to make the quotient rule easier to remember) the  the product rule should be taught as ${d(UV)=Vdu+Udv}$ This will allow students to simply change the sign on the product rule and place the difference over $V^2$ when they need to recall the quotient rule so that $${\text{while}\space d(UV)=Vdu+Udv \space \space: d(\frac {U}{V})=\frac {Vdu-Udv}{V^2}}$$

That's exactly how many of us approach the product rule, consistent with your suggested approach. 
I always teach the product rule for $\Big(f(x)g(x)\Big)'$ to be $$f'(x) g(x) + f(x) g'(x),$$ and the quotient rule  $$\left(\dfrac{f(x)}{g(x)}\right)' = \dfrac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}$$
Either presentation of the product rule is equivalent, thanks to the commutativity of addition.
So use what helps you best remember the product rule and the quotient rule.
