- Number of papers total: 45.
- Number of students who used the Quotient Rule on this exercise instead of simplifying first: 29 (64%)
- Number of students who made some kind of algebra or calculus mistake after starting the Quotient Rule: 10 (35% of those using the Quotient Rule)
On the other hand, get this: Of the 16 students who simplified the algebra first and then took the derivative, 9 of them — over half — made an algebra or calculus mistake as a result, with the most common errors being (1) multiplying by x instead of multiplying by 1/x, and (2) getting the function to equal (1/x)*(x^2 + 2x^(1/2)) but then just taking the derivative of the first factor times the derivative of the second, i.e. ignoring the Product Rule.
The first thing I learn from this is that maybe I need to rethink my warnings about the Quotient Rule making your calculations error-prone. The error rate was a lot lower for those using the Quotient Rule than it was for those doing what I said would help curb stupid algebra errors. Although my point still stands — of the errors made using the Quotient Rule, most had to do subtracting the f(x)g'(x) term and other issues that arise only because the Quotient Rule is being used; those students probably wouldn’t make algebra mistakes if they simplified first.
The second thing to notice is that despite the lower error rate for the Quotient Rule people, it still takes 4-5 times the amount of work, which if it doesn’t make you error-prone it certainly eats up more time, which in a testing/exam setting can really be deadly. So if students feel like they don’t have enough time to finish my tests (which they always do) they might want to do a careful audit of their calculation techniques and see if they aren’t contributing to the problem themselves.
The third thing to notice is that repeated exhortations about simplifying before you calculate have virtually no effect. This is a special case of the limitations of lecturing — students don’t really learn from hearing, they learn from doing. So I think they need more problems where taking the most complicated route through the problem (like here) really costs them something.