Thoughts and statistics on the Quotient Rule

The most recent calculus homework set covered basic derivative “shortcuts”, like the power rule and the quotient rule. The first exercise asked the students to find the derivative of


This is a classic “trick” exercise because it is in the section on the Power Rule, and before the section on the Quotient Rule. So the intent here is to trick students into either using a rule that doesn’t work or isn’t appropriate — like the Quotient Rule. The Quotient Rule works here but is about 4-5 times the amount of work compared to simplifying the fraction first to x – 2x^(-1/2) and then using the Power Rule. We did an example like this in class; and the next day, when we covered the Quotient Rule, I brought that example out again to say (1) yes, the Quotient Rule will work on this, but (2) it’s a lot more work and (3) the more algebra work you impose on yourself, the more error-prone you become and the more likely you are to make mistakes in the calculations. So, a life lesson emerges: always simplify the function first before invoking any calculus. Now look at the following statistics from the above exercise:

  • 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.

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Filed under Calculus, Education, Math, Teaching

4 responses to “Thoughts and statistics on the Quotient Rule

  1. Mike R.

    My (likely bad) theory is that students think that math, and homework/exam problems in particular, must be hard. They think:

    Most questions are hard, but I don’t know which. If we use the most complicated method possible, we must be doing it right. I have oversimplified problems in the past, and lost points, so I won’t assume anything and use the most complicated method possible.

    On calculus exams, if you place the questions in a different order than you taught them, you get interesting results. For example, place a derivative of a polynomial question after the product rule and quotient rule questions. Or better yet, 1/x^2. Many students assume that since the question is after the quotient rule question, you must need the quotient rule or some other more complicated procedure that they have learned. Exams always get harder, so this question must be harder. Some students try to beat the exam, not answer the questions.

  2. This morning I told to my students: “When you can explain to somebody else how to do something, and why you do that way; you´ve actually learned something. Seeing a friend when he does an exercise isn´t learning!”

  3. Dan

    What about a class/homework activity like this:

    Give students the original problem, along with its solution worked out using the two different methods (or more, if possible). Each of the solutions would have one or more errors in them. Students would need to identify the errors and correct them.

    The next section could require students to solve a similar problem using both methods. This could then be followed with a short reflection on which method they preferred and why.

    The next section could be a series of problems, some of which can be simplified and others which can’t. Students would have to identify those that can and simplify them (and maybe those that can’t be simplified and explain why).

    The final section could then call on students to actually solve a few problems from start to finish, using the method of their choice.

    Is this approach too “high school”? I can barely remember what my assignments were like when I took calculus in college.

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