Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 15 - Differentiation in Several Variables - 15.3 Partial Derivatives - Exercises - Page 782: 71


$$ g_{xyz}= 3xyz(x^2+y^2+z^2)^{-5/2}.$$

Work Step by Step

Since $ g(x,y,z)=\sqrt{x^2+y^2+z^2}$, then using the chain rule, we have $$ g_{x}=\frac{2x}{2\sqrt{x^2+y^2+z^2}}=x(x^2+y^2+z^2)^{-1/2},$$ $$ g_{xy}= -\frac{1}{2}x(x^2+y^2+z^2)^{-3/2}(2y)=-xy(x^2+y^2+z^2)^{-3/2},$$ $$ g_{xyz}=\frac{3}{2}xy(x^2+y^2+z^2)^{-5/2}(2z)=3xyz(x^2+y^2+z^2)^{-5/2}.$$
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