# The Damn Quotient Rule

Last night Kirsty and I had Tom (@tomJwicks) and his girlfriend Charlotte over for dinner. I’ve known Tom a long time – in fact since he did a summer project in the maths department at Nottingham when he was an undergraduate. He is almost at the end of his PhD and from September will be taking on the role of “Teaching Officer” in the maths department at the University of Nottingham. Part of his role will be supporting first year undergraduates in their transition to university maths, and another part will be providing enrichment type activities to local schools. In the past we have delivered many enrichment sessions together, some graduate school courses together (he took over the Introduction to Quantitative Methods and Quantitative Methods of Engineers course when I left) and various other sessions in schools before I became a teacher – it is fair to say that we have a similar style and agree on lots of things.

One of the things we have similar viewpoints on is the quotient rule that is taught at A-Level for differentiation, namely the following:

If $$f(x) = \frac{u(x)}{v(x)}$$ then $$f'(x) = \frac{u'(x)v(x) – v'(x)u(x)}{(v(x))^2}$$

If you know me, you will know that I hate the fact that this is taught… Over the course of my A-Level I realised that I made far fewer mistakes if I just used the product rule on $$u(x)v^{-1}(x)$$ directly – it seems that minus sign is pretty irritating for me! Also, I tend to find it easier to simplify expressions derived from an application of the product rule than those due to an application of the quotient rule. Personally I don’t know many people who do actually use the quotient rule anyway. My reason for not liking it being explicitly taught in the A-Level is that students often see it as another rule they need to learn, and don’t necessarily appreciate that it can just be derived from the product rule, they tend to make more sign errors too.

Tom had this to say on the quotient rule “Can’t be doing with it; Definitely should get rid of it. It has no value. In fact it’s worse than that in that it’s another thing for the students to remember and can cause a whole lot of confusion. he worst thing is when they get confused about where to put the minus sign, which clearly shows a gap in their understanding. It would just emerge if they use the product rule.”

What do you think of the product rule?

## 4 thoughts on “The Damn Quotient Rule”

1. Jemma says:

I agree. I always used the product rule myself with u(x)[v(x)]^(-1) and hated the potential for error with the quotient rule.

1. tombennison says:

Nice to see it is not just me 🙂

2. Mike Ollerton says:

I think any rule which has to be remembered is not a powerful way to learn mathematics, at any level from KS1 to undergraduate. And whilst I am not putting forward a case for the condemned (quotient rule) I offer an alternative for the learning of any rule. This is for rules and formulae to be derived which is a pedagogically sound approach to teaching mathematics per se. With this in mind I offer the following: http://mikeollerton.com/pubs/Deriving%20the%20product%20rule.pdf
Regards
Mike

1. tombennison says:

Thanks for the great comment Mike! This is the approach I like to take. I’ll write another post this week promoting the document you have linked to 😉