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I work in my university's math help center and am often presented with questions rooted in poor conceptual or intuitive understanding understanding of various mathematical questions; esp. with the beginning calculus students I work with. From day one of my calculus studies, and from theirs as well, the idea behind the derivative has always been presented with the geometric motivation of finding "the slope of the line tangent to the graph of a certain function at a particular point." But as we all know, the derivative has helped us solve many problems outside of geometry; indeed, its properties and abilities are easily adapted to many problems outside of the classic tangent line problem.

So my question is this: is there a way to construct the derivative in the context of purely algebraic problem, especially the kind of problem that can be digested by younger students? So I suppose the answer to my question is another question, one whose natural, intuitive solution involves developing a difference quotient and using it to solve the problem, in the process building the derivative.

Hopefully this might be a nice thought experiment.

Sawyer
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  • If you introduce a nilpotent who squares to zero, you can find the derivatives of polynomials purely algebraically. Whether it can be introduced to intro students I dunno. Call it an infinitesimal whose square is negligible? – ziggurism Oct 14 '16 at 02:24
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    Not sure whether this is what you have in mind, but the concept of a first-order approximation seems motivational and not geometric to me. eg consider the binomial expansion of $(a+h)^n$. When $h\ll a$ the first two terms are the most significant (a claim which can be made more rigorous using limits). – stewbasic Oct 14 '16 at 02:28
  • I'm not sure what you mean by "purely algebraic", but strictly this would exclude topological/analysis concepts such as continuity. Algebraically, the tangent can sometimes be defined without invoking continuity, for example as the (unique) line that intersects a conic at exactly one given point. This works even for conics over finite fields, though abstractly still has a geometric interpretation. Though it might not be headed in the direction you were hoping for, it carries a sort of ring of truth to it by applying without the machinery of calculus. – qman Oct 14 '16 at 05:33

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A good motivation is that the derivative of a function at some point is the best linear approximation to the function at that point.

Students are familiar with the idea that linear functions are easier to deal with than the other, more elaborate functions they've already come across. So it's easy to explain that differentiation lets you analyze even complicated functions by reducing them (in the neighborhood of any particular point) to the easier-to-understand linear case.

Mitchell Spector
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