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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