Calculus: Early Transcendentals (2nd Edition)

Published by Pearson
ISBN 10: 0321947347
ISBN 13: 978-0-32194-734-5

Chapter 3 - Derivatives - 3.4 The Product and Quotient Rules - 3.4 Exercises: 68

Answer

$f'(z)=3z^{2}e^{3z}+2z(e^{3z}+4)+\dfrac{2(z^{2}-1)}{(z^{2}+1)^{2}}$

Work Step by Step

$f(z)=z^{2}(e^{3z}+4)-\dfrac{2z}{z^{2}+1}$ Evaluate the derivative by applying the product rule on the first term and the quotient rule on the second term: $f'(z)=z^{2}(e^{3z}+4)'+(e^{3z}+4)(z^{2})'-\dfrac{(z^{2}+1)(2z)'-(2z)(z^{2}+1)'}{(z^{2}+1)^{2}}=...$ Evaluate the indicated derivatives: $...=3z^{2}e^{3z}+2z(e^{3z}+4)-\dfrac{2(z^{2}+1)-2z(2z)}{(z^{2}+1)^{2}}=...$ Simplify the fraction: $...=3z^{2}e^{3z}+2z(e^{3z}+4)-\dfrac{2z^{2}+2-4z^{2}}{(z^{2}+1)^{2}}=...$ $...=3z^{2}e^{3z}+2z(e^{3z}+4)-\dfrac{-2z^{2}+2}{(z^{2}+1)^{2}}=...$ $...=3z^{2}e^{3z}+2z(e^{3z}+4)+\dfrac{2(z^{2}-1)}{(z^{2}+1)^{2}}$
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