Understanding Equations for Beginners

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SUMMARY

This discussion focuses on understanding the differentiation of the equation represented as ##\frac{dx}{dy}\frac d{dx}\left(\frac{dy}{dx}\right)^{-1}##. The user clarifies the notation by substituting ##y'## for ##\frac{dy}{dx}##, transforming the equation into $$\frac1{y'}\frac d{dx}\left(\frac 1{y'}\right)$$. The key takeaway is the realization of the negative sign during differentiation, which is crucial for grasping the underlying mathematical principles.

PREREQUISITES
  • Understanding of basic calculus concepts, particularly differentiation.
  • Familiarity with notation such as ##\frac{dy}{dx}## and ##y'##.
  • Knowledge of inverse functions and their derivatives.
  • Ability to manipulate algebraic expressions involving derivatives.
NEXT STEPS
  • Study the rules of differentiation, focusing on the product and quotient rules.
  • Learn about inverse functions and their derivatives in calculus.
  • Practice solving differential equations to reinforce understanding.
  • Explore advanced topics in calculus, such as implicit differentiation.
USEFUL FOR

Students and educators in mathematics, particularly those beginning their studies in calculus and differentiation techniques.

funlord
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I can's understand the fact about the equation

i can't prove the equation from the first attachment to the second attachment pls help.

Sorry for bad english
 

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The second last formula is ##\frac{dx}{dy}\frac d{dx}\left(\frac{dy}{dx}\right)^{-1}##.
If we write ##y'## for ##\frac{dy}{dx}## then that is
$$\frac1{y'}\frac d{dx}\left(\frac 1{y'}\right)$$
What happens when you perform the differentiation?
 
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Thank you very much, I can understand now why it has a negative sign
 

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