Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Chapter 2 - Derivatives - 2.3 Differentiation Formulas - 2.3 Exercises - Page 141: 37


$f'(t)=\dfrac{-2t-3}{3\sqrt[3] {t^2}(t-3)^2}$

Work Step by Step

$f(t)=\dfrac{\sqrt[3] t}{t-3}$ Convert radical to exponent form: $f(t)=\dfrac{t^{1/3}}{t-3}$ Use the quotient rule: $f'(t)=\dfrac{(t-3)(\frac{1}{3}t^{-2/3})-(t^{1/3})(1)}{(t-3)^2}$ Distribute the numerator: $f'(t)=\dfrac{\frac{1}{3}t^{1/3}-t^{-2/3}-t^{1/3}}{(t-3)^2}$ Combine like terms: $f'(t)=\dfrac{-\frac{2}{3}t^{1/3}-t^{-2/3}}{(t-3)^2}$ Use the definition of a negative exponent and convert back to radical form: $f'(t)=\dfrac{-\frac{2\sqrt[3] t}{3}-\frac{1}{\sqrt[3] {t^2}}}{(t-3)^2}$ Common denominator: $f'(t)=\dfrac{\frac{-2t-3}{3\sqrt[3] {t^2}}}{(t-3)^2}$ Make into one fraction: $f'(t)=\dfrac{-2t-3}{3\sqrt[3] {t^2}(t-3)^2}$
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