Calculus 8th Edition

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

Chapter 6 - Inverse Functions - 6.3* The Natural Exponential Function - 6.3* Exercises: 40

Answer

$-\frac{(t+2)^2}{t^2e^t}$

Work Step by Step

$\frac{d}{dt}\frac{4+t}{te^t}$ $=\frac{te^t\frac{d}{dt}(4+t)-(4+t)\frac{d}{dt}(te^t)}{(te^t)^2}$ $=\frac{te^t*1-(4+t)(t\frac{d}{dt}e^t+e^t\frac{d}{dt}t)}{(te^t)^2}$ $=\frac{te^t-(4+t)(te^t+e^t*1)}{(te^t)^2}$ $=\frac{te^t-e^t(4+t)(t+1)}{(te^t)^2}$ $=\frac{e^t(t-(t^2+5t+4))}{e^t(t^2e^t)}$ $=-\frac{t^2+4t+4}{t^2e^t}$ $=-\frac{(t+2)^2}{t^2e^t}$
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