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\begin{document}
\begin{center}
\Huge{
Differentiation Practice}
\end{center}

\begin{enumerate}
\item Find $\displaystyle{\frac{dy}{dx}}$ for each of the following  \emph{and simplify}:

\hspace{-.5in}$y=\ln(x)+2\log(x)$ \hspace{1in} $y=5\log_2(x^3+2)$\hspace{1in} $y=\ln\left[4x-3\right]+\ln \left[\sin(x)\right]$

\vfill
\item Find $\displaystyle{\frac{dy}{dx}}$ for each of the following \emph{and simplify}:

\hspace{-.5in}$y=\ln(x)+\log(x^2)$ \hspace{1in} $y=\log_2\left[(x^3+2)^5\right]$\hspace{1in} $y=\ln\left[(4x-3)\sin(x)\right]$
\vspace{2.3in}
\item Notice any similarities between the answers to 1 and the answers to 2?  
\vspace{.2in}
\item Is there any precalculus/algebra tools you can use to modify the problems in 1 to get the problems in 2?
\vspace{.2in}

If you don't remember your logarithm properties from precalculus check out Appendix G on page A51 and try to reanswer the above question.
\end{enumerate}

\newpage

\begin{center}
\Huge{Logarithmic Differentiation }
\end{center}


\normalsize

~\\
\noindent
Let $b$ be a positive real number.  Recall the properties of logarithms:

\begin{tabular*}{6in}{l@{\extracolsep{1.5in}}l}
$\displaystyle{\log_b(xy)=\log_b x+ \log_b y}$\\
$\displaystyle{\log_b\left(\frac{x}{y}\right)=\log_b x-\log_b y}$\\
$\displaystyle{\log_b(x^y)=y\log_bx}$\\
\end{tabular*}

\noindent
Note: you need to \emph{know} these for quizzes and exams as they will not be provided for you!!!

\begin{enumerate}
\item Use any method you like to find $\frac{dy}{dx}$.

$\displaystyle{y=\sqrt{\frac{x-1}{x^4+1}}}$ \hspace{3in}$y=x^{\sqrt{x}}$
\vspace{2.5in}

\item Let $y=(\sin x)^{\ln x}$.  
\begin{itemize}
\item True or False: $\frac{dy}{dx}=\ln(x)(\sin x)^{\ln x-1}$.
\vskip .5in
\item Use logarithmic differentiation to find $\frac{dy}{dx}$.
\end{itemize}

\end{enumerate}

\end{document}
