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: 39


$$ Q_L= \left(\frac{1}{M}-\frac{tL}{M^2}\right)e^{-Lt/M},\\ Q_M =-\left(\frac{L}{M^2}-\frac{L^2t}{M^3}\right)e^{-Lt/M},\\ Q_t=-\frac{L^2}{M^2}e^{-Lt/M}. $$

Work Step by Step

Recall the product rule: $(uv)'=u'v+uv'$ Recall that $(e^x)'=e^x$ Since $ Q=\frac{L}{M}e^{-Lt/M}$, then by the product rule, we have $$ Q_L= \frac{1}{M}e^{-Lt/M}+ \frac{L}{M}e^{-Lt/M}\frac{-t}{M}=\left(\frac{1}{M}-\frac{tL}{M^2}\right)e^{-Lt/M},\\ Q_M= - \frac{L}{M^2}e^{-Lt/M}+ \frac{L}{M}e^{-Lt/M}\frac{Lt}{M^2}=-\left(\frac{L}{M^2}-\frac{L^2t}{M^3}\right)e^{-Lt/M},\\ Q_t=-\frac{L^2}{M^2}e^{-Lt/M}. $$
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