%% Math 131, Fall, 2003
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\begin{document}

\begin{center}
\textbf{Holy Cross College, Fall Semester, 2003} \\
\textbf{Math 131, Review Sheet for Midterm 2} \\
Selected Solutions \\
Professor Hwang \\
\end{center}

\be

\item Find the indicated derivatives; you may want to perform algebra
  before differentiating:

  \bd

  \item{(a)} $\ddx{u}\frac{1}{\ln u}=-\frac{1}{u(\ln u)^2}$, \quad
    $\ddx{x}x\ln x=1+\ln x$, \quad 
    $\ddx{t}\ln(\ln t)=\frac{1}{t\ln t}$,\\
    $\ddx{x}\ln\frac{x-1}{x+1}=\frac{2}{x^2-1}$ 

  \item{(b)} $\ddx{x}\sqrt{x^2+1}=\frac{x}{\sqrt{x^2+1}}$,\qquad
  $\ddx{v}\sqrt{e^v+e^{-v}}
    =\frac{1}{2}\frac{e^v-e^{-v}}{\sqrt{e^v+e^{-v}}}$

  \item{(c)}
  $\ddx{s}\frac{\sqrt{s}}{s^2+1}=\frac{1-3s^2}{2(s^2+1)^2\sqrt{s}}$,
  \qquad $\ddx{w}\frac{w^2-1}{w-1}=1$ 

  \ed

\ee

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