Thomas' Calculus 13th Edition

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

Chapter 14: Partial Derivatives - Section 14.3 - Partial Derivatives - Exercises 14.3 - Page 807: 27

Answer

$f_x=\frac{yz}{\sqrt (1-x^2y^2z^2)}$ $f_y=\frac{xz}{\sqrt (1-x^2y^2z^2)}$ $f_z=\frac{xy}{\sqrt (1-x^2y^2z^2)}$

Work Step by Step

Take the first partial derivatives of the given function. When taking a partial derivative with respect to x, treat y and z as constants, with respect to y, treat x and z as constants, and with respect to z, treat x and y as constants: $f_x=\frac{1}{\sqrt (1-(xyz)^2)}\times yz=\frac{yz}{\sqrt (1-x^2y^2z^2)}$ $f_y=\frac{1}{\sqrt (1-(xyz)^2)}\times xz=\frac{xz}{\sqrt (1-x^2y^2z^2)}$ $f_z=\frac{1}{\sqrt (1-(xyz)^2)}\times xy=\frac{xy}{\sqrt (1-x^2y^2z^2)}$
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