Not to be a calculus heretic or anything, but…

…can anybody give me some justification for teaching implicit differentiation and related rates in my Calculus I course? If not, I’m going to drop them from the syllabus.

Of course, I’ve been teaching these topics since forever, but in thinking about content getting in the way of understanding, I’ve been wondering what I might cut from my calculus courses to create time which I can reinvest on more foundational things. These two topics rose immediately to the top of my list; they require a high investment in time and effort and seem to have an awfully low yield of understanding, no matter how much you invest. Is it worth it to keep teaching them?

I can understand including implicit differentiation, because we use this technique to get derivative formulas for inverse functions (like logarithms). But can’t you just call it the Chain Rule instead? Do you have to go through all the rigmarole of finding tangent lines to curves that aren’t functions just to get d/dx(ln(x)) = 1/x, which will then be promptly misremembered? (I should note that our Calc I doesn’t include any trigonometry — that’s saved for Calc II — so we don’t fool with inverse trig functions. So implicit differentiation is used by us to derive exactly one derivative formula. Again, is it worth it?) 

Not that I think these topics are unimportant, but my students are failing to get some really fundamental and useful concepts from calculus (like, the meaning of a derivative in the first place) that are a lot further down the concept food chain than these, and we need the time and space to work harder on them. I don’t want to include topics on a syllabus just for the sake of including them.

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One response to “Not to be a calculus heretic or anything, but…

  1. When I teach implicit, I call it “chain rule.” I don’t make a big fuss about it.

    For both these topics, I think that their value comes as part of the bigger picture of the students’ programs. Most of my calculus students are engineering students and will need to take Calc 1-2-3 plus physics. The implicit differentiation warms them up for some of the things they’ll see in Calc 3 (like deciding what’s an independent variable and what’s a function). And I think that physics expects them to know it.

    The only value that I see to related rates is that it’s more practice with word problems — something that my students definitely need. I would happily skip it if I knew how to productively fill the time.