How come I’m always stuffing it up when I use the quotient rule? Product rule is ok. They’re pretty much the same, right?
Well, yes and no. They are very similar. Let’s have a look:
They are similar but the Product Rule has much more ‘give’ in it. It is much more forgiving of a casual approach to the problem. The reason for that goes right back to the properties of the basic mathematical operations of addition, subtraction, multiplication, and division.
Notice in the Product Rule that there is only addition and multiplication. Addition and multiplication are both commutative. That means the order in which you add or multiply doesn’t matter. For example:
So for the Product Rule, provided all the individual elements are correct, you can just about put them together in whatever order you like. It’s kinda hard to get it wrong in a way.
That is definitely not the case for the Quotient Rule. You’ve got subtraction and division in there and neither of those are commutative. Order is important when subtracting or dividing. For example:
Because of that you have to be very mindful of the order in which you put the elements together when using the Quotient Rule.
Students are usually, and rightly, introduced to the Product Rule first because it is easier. But sometimes students get into the habit of approaching Product Rule problems casually, and because they do look similar, carry that attitude into Quotient Rule problems.
Be careful. Respect the Quotient 