Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Chapter 14 - Partial Derivatives - 14.4 Tangent Planes and Linear Approximations - 14.4 Exercises - Page 975: 25


$dz=-2e^{-2x}cos(2 \pi t) dx -2\sin(2 \pi t)e^{-2x} dt$

Work Step by Step

The differential is defined as: $dw=\frac{\partial w}{\partial x} dx + \frac{\partial w}{\partial y} dy + \frac{\partial w}{\partial z} dz$ Given the function $z=e^{-2x}cos(2 \pi t)$ . Thus, we find the partial derivatives w.r.t $t$ and $x$: $f_x=-2e^{-2x}cos(2 \pi t)$ $f_t=-2\sin(2 \pi t)e^{-2x}$ Thus, the differential: $dz=-2e^{-2x}cos(2 \pi t) dx -2\sin(2 \pi t)e^{-2x} dt$
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