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: 33


$$ U_r =(\frac{-1}{r^2}-\frac{t}{r})e^{-rt},\quad U_t= -e^{-rt}. $$

Work Step by Step

Recall the product rule: $(uv)'=u'v+uv'$ Recall that $(e^x)'=e^x$ Since $ U=e^{-rt}/r=r^{-1}e^{-rt}$, then by using the product rule, we have $$ U_r=-r^{-2}e^{-rt}+r^{-1}e^{-rt}(-t)=(\frac{-1}{r^2}-\frac{t}{r})e^{-rt},\\ U_t= r^{-1}e^{-rt}(-r)=-e^{-rt}. $$
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