B Why Can Derivatives Be Treated Like Fractions in Solving Equations?

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Derivatives can be treated like fractions in solving equations due to their origins in the concept of limits, where the derivative represents a ratio of small changes (dy/dx). This treatment is often justified through the Chain Rule, but caution is advised as not all manipulations of derivatives are valid. The notation of differentials, such as dy and dx, serves to indicate the variable of integration and reflects the underlying calculus principles. While some educational approaches, like using hyperreals, allow for treating derivatives as fractions more freely, traditional calculus emphasizes the importance of limits for a solid foundation. Understanding these concepts helps clarify why derivatives can be manipulated similarly to fractions in certain contexts.
  • #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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