How to use the Quotient Rule to Find Both First Order Partial Derivatives of f(x, y) = xy/(x + y) | Summary and Q&A

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November 1, 2020
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How to use the Quotient Rule to Find Both First Order Partial Derivatives of f(x, y) = xy/(x + y)

TL;DR

Learn how to find partial derivatives using the quotient rule in calculus with step-by-step examples.

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Key Insights

  • 🤝 The quotient rule is a fundamental concept in calculus used to find derivatives when dealing with fractions of functions.
  • 📏 When calculating partial derivatives, treat other variables as constants and apply the quotient rule accordingly.
  • 😨 The process of finding partial derivatives requires attention to detail and care in differentiating specific variables.
  • 📏 Understanding how to apply the quotient rule can simplify the process of finding partial derivatives in calculus.
  • ❓ The final answers for partial derivatives showcase the impact of variables and constants on the differentiation process.
  • 📏 Practice and repetition can enhance proficiency in finding partial derivatives using techniques like the quotient rule.
  • 🫡 Calculating partial derivatives with respect to different variables provides insight into how functions change based on specific inputs.

Transcript

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Questions & Answers

Q: What is the quotient rule in calculus and how is it used to find partial derivatives?

The quotient rule states that the derivative of a quotient of two functions is the derivative of the numerator times the denominator minus the numerator times the derivative of the denominator, all over the denominator squared. In the context of partial derivatives, we apply this rule when finding the derivative of a function with respect to one variable while treating the other variables as constants.

Q: How do you calculate the partial derivative of a function with respect to x?

To find the partial derivative of a function with respect to x, treat all other variables as constants. Apply the quotient rule by taking the derivative of the numerator with respect to x, leaving the other variables as is, and then subtracting the product of the numerator and the derivative of the denominator with respect to x. Finally, divide the result by the denominator squared to get the partial derivative.

Q: What is the process for calculating the partial derivative of a function with respect to y?

When finding the partial derivative of a function with respect to y, all other variables (such as x) are considered constants. Use the quotient rule by differentiating the numerator with respect to y and then subtracting the product of the numerator and the derivative of the denominator with respect to y. Divide the result by the denominator squared to obtain the partial derivative.

Q: How are the final answers for the partial derivatives expressed in the provided examples?

The final answers for the partial derivatives are expressed in simplified forms, taking into account the calculations done using the quotient rule. The results show the relationship between the variables present in the original function.

Summary & Key Takeaways

  • Explanation of the quotient rule for finding partial derivatives in calculus.

  • Step-by-step calculation of partial derivatives with respect to x and y.

  • Final answers provided for each partial derivative.

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