Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 7 - Exponential Functions - 7.1 Derivative of f(x)=bx and the Number e - Exercises - Page 328: 67


$$k(b-a) \left( b \right)^{\frac{1}{m}-1}$$

Work Step by Step

Given $$M(t)=\left(a+(b-a) e^{k m t}\right)^{1 / m}$$ Since \begin{align*} M^{\prime}(t)&=\frac{1}{m}\left(a+(b-a) e^{k m t}\right)^{\frac{1}{m}-1}\left(k m(b-a) e^{k m t}\right)\\ &=k(b-a) e^{k m t}\left(a+(b-a) e^{k m t}\right)^{\frac{1}{m}-1} \end{align*} Then \begin{align*} M^{\prime}(0) &=k(b-a) e^{k m (0)}\left(a+(b-a) e^{k m (0)}\right)^{\frac{1}{m}-1}\\ &= k(b-a) \left( b \right)^{\frac{1}{m}-1} \end{align*}
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