Convention verses memory: The quotient rule v product rule for derivatives

Question

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 $${\text{while}\space d(UV)=Vdu+Udv \space \space: d(\frac {U}{V})=\frac {Vdu-Udv}{V^2}}$$

Answer 1

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.

Answer 2

Probably the answer is just that nobody really cared.

Anyway, it doesn't seem to me so much better that way;

With the product rule, you just have to remember Derivative * Non derivative + Non derivative * Derivative is whatever order you like.

You don't focus on that, so it is easier and faster.

With the quotient rule order matters, and students are gonna learn it.

If we would focus on the order on product rule, students will still have to remeber some ordering, and they would have a bigger chance at getting confused (the minus sign is on the product rule or on the quotient rule? )