#### Answer

\[{s^,} = t + 2t\ln \left| t \right|\,\]

#### Work Step by Step

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