Product rule instead of quotient rule?

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SUMMARY

The discussion centers on the differentiation of the function x/(x²+1) using alternative methods. Participants confirm that it is valid to rewrite the function as x * (x²+1)-1 and apply the product and chain rules instead of the quotient rule. This approach is not only permissible but can also simplify the differentiation process. Additionally, the equivalence of the two methods is highlighted, suggesting that deriving the quotient rule from this perspective is a worthwhile exercise.

PREREQUISITES
  • Understanding of basic calculus concepts, specifically differentiation.
  • Familiarity with the quotient rule for derivatives.
  • Knowledge of the product and chain rules in differentiation.
  • Ability to manipulate algebraic expressions involving exponents.
NEXT STEPS
  • Explore the derivation of the quotient rule from first principles.
  • Practice differentiating various functions using both the product and quotient rules.
  • Study the implications of using different differentiation techniques on function simplification.
  • Learn about advanced differentiation techniques, such as implicit differentiation.
USEFUL FOR

Students of calculus, mathematics educators, and anyone looking to deepen their understanding of differentiation techniques and their applications.

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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