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: 14

Answer

\[{y^,} = \frac{{ - 2{x^2}}}{{2 - {x^2}}} + \ln \left| {2 - {x^2}} \right|\]

Work Step by Step

\[\begin{gathered} y = x\ln \left| {2 - {x^2}} \right| \hfill \\ Differentiate \hfill \\ {y^,} = \,\,{\left[ {x\ln \left| {2 - {x^2}} \right|} \right]^,} \hfill \\ Use\,\,the\,\,product\,\,rule \hfill \\ {y^,} = x\,\,{\left[ {\ln \left| {2 - {x^2}} \right|} \right]^,} + \ln \left| {2 - {x^2}} \right|\,{\left( x \right)^,} \hfill \\ Use\,\,\frac{d}{{dx}}\,\,\left[ {\ln g\,\left( x \right)} \right] = \frac{{{g^,}\,\left( x \right)}}{{g\,\left( x \right)}} \hfill \\ {y^,} = x\,\left( {\frac{{\,{{\left( {2 - {x^2}} \right)}^,}}}{{2 - {x^2}}}} \right) + \ln \left| {2 - {x^2}} \right|\,\left( 1 \right) \hfill \\ {y^,} = x\,\left( {\frac{{\, - 2x}}{{2 - {x^2}}}} \right) + \ln \left| {2 - {x^2}} \right| \hfill \\ Multiply \hfill \\ {y^,} = \frac{{ - 2{x^2}}}{{2 - {x^2}}} + \ln \left| {2 - {x^2}} \right| \hfill \\ \end{gathered} \]
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