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: 43


$g_{xx}=2y-ysinx$ $g_{yy}=-cosy$ $g_{xy}=g_{yx}=2x+cosx$

Work Step by Step

Take the first partial derivatives of the given function. When taking partial derivative with respect to x, treat y as a constant, and vice versa: $g_x=2xy+ycosx$ $g_y=x^2-siny+sinx$ Then take the derivative of the first order partial derivatives to find second partial derivatives: $g_{xx}=2y-ysinx$ $g_{yy}=-cosy$ Second partial derivatives of first order partial derivatives of x with respect to y and y with respect to x are the same: $g_{xy}=g_{yx}=2x+cosx$
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