Why Can Derivatives Be Treated Like Fractions in Solving Equations?

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    Derivatives Fractions
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Discussion Overview

The discussion revolves around the treatment of derivatives as fractions in solving equations, exploring the reasoning behind this practice and its implications. Participants examine the conceptual and mathematical foundations of derivatives, differentials, and integration, as well as the potential pitfalls of treating derivatives like fractions.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants express confusion about why derivatives can be treated like fractions, questioning the logic behind this practice.
  • Others explain that derivatives originated from the concept of slopes, which are based on ratios of small changes (dy/dx), and that this fraction-like treatment is sometimes justified through the use of differentials.
  • One participant notes that while the Chain Rule allows for certain fraction-like manipulations, there are limits to this analogy, as some manipulations are invalid.
  • Some contributions discuss the use of hyperreals in calculus, suggesting that this framework allows for treating dy/dx as a fraction more comfortably, although others caution that this approach may not be suitable for all students.
  • Participants debate the significance of the differential (dx) in integration, with some arguing that it represents the smallest change in a variable and is essential for understanding the process of integration.
  • There are discussions about the validity of integrating expressions involving higher-order derivatives and the importance of maintaining proper notation in calculus.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the treatment of derivatives as fractions. There are multiple competing views regarding the validity and implications of this practice, as well as differing opinions on the educational approaches to teaching calculus.

Contextual Notes

Some participants highlight limitations in understanding the foundational concepts of calculus, particularly regarding the role of limits and the mathematical rigor behind treating derivatives as fractions. There are unresolved questions about the implications of using hyperreals and the appropriateness of various teaching methods.

  • #31
Dale said:
In a calculus class I would teach only as much real analysis or sequences as is absolutely necessary to teach calculus. I think that is pretty minimal...
I think that would not be wise. Engineers and applied mathematicians would probably be hurt by that.
 
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  • #32
FactChecker said:
I think that would not be wise. Engineers and applied mathematicians would probably be hurt by that.
How?
 
  • #33
Dale said:
How?
Engineers and applied mathematicians need to be very experienced and comfortable with convergent sequences and series of all sorts.
 

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