Product rule instead of quotient rule?

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Discussion Overview

The discussion revolves around the differentiation of the function x/(x²+1) and whether it is permissible to use the product rule instead of the quotient rule by rewriting the expression. The scope includes mathematical reasoning related to calculus techniques.

Discussion Character

  • Mathematical reasoning

Main Points Raised

  • One participant suggests using the product rule by rewriting the function as (x)(x²+1)-1.
  • Another participant agrees that this approach is valid.
  • A different participant proposes deriving the quotient rule using this method as an exercise.
  • One participant asserts that both methods yield the same result and suggests that using the product rule may be easier if simplification is not required.

Areas of Agreement / Disagreement

Participants generally agree that using the product rule in this context is permissible and that both methods are equivalent, although the discussion does not delve into any disagreements.

Flatland
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Say for example, to differentiate x/(x²+1) I would use to quotient rule. However, would it be legal to bring up the denominator to: (x)(x²+1)-¹ and use the product/chain rule instead?
 
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Yes.....
 
Sure. As an exercise, try deriving the quotient rule using that trick.
 
yes, they are the same. if you need not simplify, i think that way is even easier.
 

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