Calculus with Applications (10th Edition)

Published by Pearson
ISBN 10: 0321749006
ISBN 13: 978-0-32174-900-0

Chapter 4 - Calculating the Derivative - 4.5 Derivatives of Logarithmic Functions - 4.5 Exercises - Page 240: 19


\[{y^,} = \frac{{6x\ln x - 3x}}{{\,{{\left( {\ln x} \right)}^2}}}\]

Work Step by Step

\[\begin{gathered} y = \frac{{3{x^2}}}{{\ln x}} \hfill \\ Differentiate \hfill \\ {y^,} = \,\,{\left[ {\frac{{3{x^2}}}{{\ln x}}} \right]^,} \hfill \\ Use\,\,the\,\,quotient\,\,rule \hfill \\ {y^,} = \frac{{\,\left( {\ln x} \right)\,{{\left( {3{x^2}} \right)}^,} - \,\left( {3{x^2}} \right)\,{{\left( {\ln x} \right)}^,}}}{{\,{{\left( {\ln x} \right)}^2}}} \hfill \\ Where\,\,\,\,{\left[ {\ln x} \right]^,} = \frac{1}{x}\,\,,\,\,Then \hfill \\ {y^,} = \frac{{\,\left( {\ln x} \right)\,\left( {6x} \right) - 3{x^2}\,\left( {\frac{1}{x}} \right)}}{{\,{{\left( {\ln x} \right)}^2}}} \hfill \\ Multiplying \hfill \\ {y^,} = \frac{{6x\ln x - 3x}}{{\,{{\left( {\ln x} \right)}^2}}} \hfill \\ \end{gathered} \]
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