Calculus, 10th Edition (Anton)

Published by Wiley
ISBN 10: 0-47064-772-8
ISBN 13: 978-0-47064-772-1

Chapter 13 - Partial Derivatives - 13.5 The Chain Rule - Exercises Set 13.5 - Page 959: 71

Answer

See explanation

Work Step by Step

Work Step by Step Step 1 In this problem, we will compare the process of implicit differentiation with the use of the following formula: \[ \frac{dy}{dx} = -\frac{\partial f / \partial x}{\partial f / \partial y} \] This formula is based on the relationship between partial derivatives and can be applied when we have the equation of the form $f(x, y) = c$. Essentially, we are performing implicit differentiation using this formula. Step 2 Let's delve into the background of the formula: \[ \frac{dy}{dx} = -\frac{\partial f / \partial x}{\partial f / \partial y} \] We can use this formula when dealing with special cases where $ f(x, y) = c$. In such cases, implicit differentiation becomes more manageable by employing this formula. Step 3 In a more general case of implicit differentiation, we differentiate every term with respect to the independent variable (typically $x$). Afterward, we use the chain rule to simplify further. Once we've simplified each term, we proceed to solve for the desired derivative by rearranging the terms and isolating the derivative.
Update this answer!

You can help us out by revising, improving and updating this answer.

Update this answer

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.