Thomas' Calculus 13th Edition

Published by Pearson
ISBN 10: 0-32187-896-5
ISBN 13: 978-0-32187-896-0

Chapter 14: Partial Derivatives - Practice Exercises - Page 864: 20


$f_{x}=\frac{x-y}{x^{2}+y^{2}}$ $f_{y}=\frac{x+y}{x^{2}+y^{2}}$

Work Step by Step

$f(x,y)=\frac{1}{2}\ln(x^{2}+y^{2})+tan^{-1}\frac{y}{x}$ $f_{x}=\frac{1}{2}\frac{2x}{x^{2}+y^{2}}+\frac{-\frac{y}{x^2}}{1+(\frac{y}{x})^2}=\frac{x}{x^{2}+y^{2}}-\frac{y}{x^{2}+y^{2}}=\frac{x-y}{x^{2}+y^{2}}$ $f_{y}=\frac{1}{2}\frac{2y}{x^{2}+y^{2}}+\frac{\frac{1}{x}}{1+(\frac{y}{x})^2}=\frac{x+y}{x^{2}+y^{2}}$
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