Calculus: Early Transcendentals 8th Edition

Published by Cengage Learning
ISBN 10: 1285741552
ISBN 13: 978-1-28574-155-0

Chapter 3 - Review - Exercises: 36

Answer

$y'=\dfrac{4+\ln(t^{4})}{2\sqrt{t\ln(t^{4})}}$

Work Step by Step

$y=\sqrt{t\ln(t^{4})}$ Rewrite the square root using a rational exponent: $y=[t\ln(t^{4})]^{1/2}$ Start the differentiation process by using the chain rule: $y'=\dfrac{1}{2}[t\ln(t^{4})]^{-1/2}[t\ln(t^{4})]'=...$ Change the sign of the negative exponent by changing its corresponding factor to the denominator: $...=\dfrac{[t\ln(t^{4})]'}{2[t\ln(t^{4})]^{1/2}}=...$ Rewrite the rational exponent in the denominator as a square root: $...=\dfrac{[t\ln(t^{4})]'}{2\sqrt{t\ln(t^{4})}}$ Use the product rule to evaluate the derivative indicated in the numerator: $...=\dfrac{t[\ln(t^{4})]'+(t)'\ln(t^{4})}{2\sqrt{t\ln(t^{4})}}=...$ Use the chain rule one last time to evaluate the derivatives indicated in the numerator and simplify: $...=\dfrac{t\Big[\dfrac{(t^{4})'}{t^{4}}\Big]+\ln(t^{4})}{2\sqrt{t\ln(t^{4})}}=\dfrac{t\Big(\dfrac{4t^{3}}{t^{4}}\Big)+\ln(t^{4})}{2\sqrt{t\ln(t^{4})}}=\dfrac{4+\ln(t^{4})}{2\sqrt{t\ln(t^{4})}}$
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