Double derivative of a quotient

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SUMMARY

The discussion focuses on finding the second derivative of a quotient, specifically d²y/dx², where dy/dx is given as (3x² - 24x - 45) ÷ 2y. Participants emphasize the use of the product rule and chain rule for differentiation, noting the necessity of substituting dy/dx back into the equation. Additionally, they highlight the importance of determining y² through integration using separation of variables to complete the solution.

PREREQUISITES
  • Understanding of calculus, specifically differentiation and integration
  • Familiarity with the product rule and chain rule in calculus
  • Knowledge of separation of variables for solving differential equations
  • Basic algebraic manipulation skills
NEXT STEPS
  • Study the application of the product rule in calculus
  • Learn about the chain rule and its implications in differentiation
  • Explore separation of variables for solving first-order differential equations
  • Investigate techniques for integrating functions involving variables in the denominator
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Students and professionals in mathematics, particularly those studying calculus, as well as educators looking to enhance their understanding of derivatives and integration techniques.

chung963
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Find d^2y/dx^2 when dy/dx = (3x^2 - 24x - 45) ÷ 2y

i tried by (6x-24) ÷ 2y. Unsure what to do about the y on the denominator.
 
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dy/dx = (3x2-24x-45) * (2y)-1

Use the product rule and the chain rule (the derivative of y is dy/dx). After you have that, make a substitution for dy/dx, since you already have that. Of course, you'll also need y2, so you also need to integrate using separation of variables to find what y2 is.
 

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