How Do Differentials Apply to Implicit Functions in Calculus?

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SUMMARY

The discussion centers on the application of differentials to implicit functions in calculus, emphasizing the concept of infinitesimals. Participants highlight that while differentials can be viewed as fractions of infinitely small numbers, this leads to a contradiction since all real numbers are finite. The conversation clarifies that infinitesimals, defined as non-real numbers satisfying the condition 0<ε PREREQUISITES

  • Understanding of basic calculus concepts, particularly derivatives and differentials.
  • Familiarity with implicit functions and their properties.
  • Knowledge of infinitesimals and their role in calculus.
  • Basic grasp of real and non-real numbers in mathematical contexts.
NEXT STEPS
  • Research the application of infinitesimals in calculus, focusing on non-standard analysis.
  • Study the properties and applications of implicit differentiation in calculus.
  • Explore the concept of limits and their relationship with differentials.
  • Learn about the historical development of calculus and the introduction of infinitesimals.
USEFUL FOR

Students of calculus, mathematics educators, and anyone interested in advanced mathematical concepts related to differentials and implicit functions.

Timothy S
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I have recently come across the use of differentials in visualizing and thinking about calculus. In this method, one thinks of dx/dy as an actual fraction of infinity small yet real numbers. How is it possible to apply this to implicit functions?
 
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Timothy S said:
infinity small yet real numbers

Contradiction in terms. Every real number is finite by definition. What you can do is consider non-real numbers that satisfy ##0<\epsilon<r## for every real number r. This is the core of infinitesimals.
 

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