If we accept that there's not much learning from doing the "same" questions, like find the derivative of $x^2$, and $x^3$, and $x^4$ due to the algorithmic way of how it's done, then what happens when it applies to questions in a more subtle way?

To use an example, I've always found implicit differentiation to be very algorithmic, to the point that comparing different examples will showcase the same general ideas but with different functions. For example, if you look at example 5 of Paul's online math notes, aren't those 3 specific examples are structurally the same, but have variations of functions. I could have given 3 steps in the algorithm as this:

  1. Differentiate with respect to x

  2. Remember to use chain rule for y so that you get y'

  3. Solve for the derivative y'

For sake of argument, if I'm designing questions on implicit differentiation to test if students know a calculus concept, rather than designing a test to see if they know how to algebraically manipulate things, then (1) should I do more creative questions, and (2) how would I do that?

I think open ended questions and examples would be more helpful for conceptual understanding, like "at what angles do these curves intersect?" or "are these curves orthogonal?" but I won't specifically mention implicit differentiation. It might be likely that the answer is found via implicit differentiation if it's in the implicit differentiation section, but don't think I can do much about that. Or maybe "what's the relationship between $c$, and the $x$- and $y$-intercepts of any line tangent to the graph of $\sqrt{x}+\sqrt{y}=\sqrt{c}$?"

But anyway, in general, if I analyze a bunch of questions and see that they're all variations of the same structure, then should that structure be pointed out? (I find, if it is, then we can move onto more interesting and/or creative questions. Or, at least, move onto questions that focus on understanding the concepts at hand rather than seeing if the student knows the prerequisite algebraic manipulation skills)

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    $\begingroup$ "If we accept that there's not much learning from doing the "same" questions, like find the derivative of x2, and x3, and x4 due to the algorithmic way of how it's done..." But I don't accept that. Something new, needs repetition. (Look at all the complaints we get about kids who can't combine fractions!) $\endgroup$
    – guest
    Oct 20 '20 at 11:42
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    $\begingroup$ @guest I know what you mean, but isn't the point of the examples and tests about focusing in on the the concepts of what's being taught. Calculations are going to be showcased no matter the question, but if I want to showcase concepts, wouldn't I have to look beyond calculations? We also seem to have a difference of what's needed for a class since, if a class is on implicit differentiation, then that's the concepts that needs to be explored and not just the algebraic manipulation techniques. $\endgroup$
    – user13544
    Oct 21 '20 at 1:34
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    $\begingroup$ It seems to me that this is extremely course and student dependent. In some cases, just being able to carry out the mechanics of implicit differentiation for something like $x^3y + 5xy^2 = 4x^2y^3 + 2$ (along with "applications", such as finding the equation of the tangent line to $x^3y + 5xy^2 = 4x^2y^3 + 2$ at the point $(2,1))$ is all that's expected (e.g. a 1-semester "calculus for life sciences" course), and in other cases what you're desiring and more is expected (e.g. honors calculus courses using Spivak's book). $\endgroup$ Oct 21 '20 at 16:15
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    $\begingroup$ Incidentally, it's usually obvious that implicit differentiation is needed in textbook examples that appear after the book's coverage of implicit differentiation. I used to give examples of functions in which, after clearing fractions and radicals (when easily done), one is left with an implicit equation from which it is much easier to find $y'$ than was the case with the original equation. You can also use it to show how to (formally, in the sense of manipulation) find the derivatives of specific inverse functions (e.g. $\arcsin x)$ before showing how the method works in general. $\endgroup$ Oct 21 '20 at 16:25
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    $\begingroup$ Teaching Tip: Here's how I came up with the example that passes through $(2,1)$ a couple of comments ago. First, expand $(x-1)(x-2)(x-3)$ as $x^3 - 4x^2 + 5x - 2.$ Thus, $x=2$ is a solution of $x^3 + 5x = 4x^2 + 2.$ Next, throw on some $y$'s, such as $x^3y + 5xy^2 = 4x^2y^3 + 2,$ so that when $y=1$ we obtain the previous equation. Now we have an equation that's cubic in each of $x$ and $y$ (and so probably not easily solvable for either in terms of the other) whose graph contains the point $(2,1).$ With a little more work, you can get such curves to pass through points like $(\sqrt{2}, 5).$ $\endgroup$ Oct 21 '20 at 16:33

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