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 781: 40


$$ w_x =\frac{1}{(x^2+y^2+z^2)^{3/2}}-\frac{3x^2}{(x^2+y^2+z^2)^{5/2}}, \\ w_y= -\frac{3xy}{(x^2+y^2+z^2)^{5/2}}, \\ w_z= -\frac{3xz}{(x^2+y^2+z^2)^{5/2}}. $$

Work Step by Step

Since $ w=\frac{x}{(x^2+y^2+z^2)^{3/2}}= x(x^2+y^2+z^2)^{-3/2}$, then using the product and chain rules, we have $$ w_x= (x^2+y^2+z^2)^{-3/2}-\frac{3}{2} x(x^2+y^2+z^2)^{-5/2}(2x)=\frac{1}{(x^2+y^2+z^2)^{3/2}}-\frac{3x^2}{(x^2+y^2+z^2)^{5/2}}, \\ w_y= -\frac{3}{2} x(x^2+y^2+z^2)^{-5/2}(2y)= -\frac{3xy}{(x^2+y^2+z^2)^{5/2}}, \\ w_z= -\frac{3}{2} x(x^2+y^2+z^2)^{-5/2}(2z)= -\frac{3xz}{(x^2+y^2+z^2)^{5/2}}. $$
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