Calculus 10th Edition

Published by Brooks Cole
ISBN 10: 1-28505-709-0
ISBN 13: 978-1-28505-709-5

Chapter 13 - Functions of Several Variables - 13.4 Exercises - Page 905: 10

Answer

\begin{aligned} d w= 2 x y z^{2} d x+\left(x^{2} z^{2}+z \cos y z\right) d y +\left(2 x^{2} y z+y \cos y z\right) d z \end{aligned}

Work Step by Step

Given $$ w = x^{2} y z^{2}+\sin (y z)$$ Since $$dw=\frac{\partial w}{\partial x} dx+\frac{\partial w}{\partial y} dy+\frac{\partial w}{\partial z} dz,$$ $$\frac{\partial w}{\partial x} =2 x y z^{2} ,$$ $$\frac{\partial w}{\partial y} =x^{2} z^{2}+ z\cos (y z)$$ and $$\frac{\partial w}{\partial z} =2x^{2} y z+y \cos (y z) $$ then we get \begin{aligned} d w= 2 x y z^{2} d x+\left(x^{2} z^{2}+z \cos y z\right) d y +\left(2 x^{2} y z+y \cos y z\right) d z \end{aligned}
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