\hfill \hfill } %} \newtheorem{theorem}{Theorem} \newtheorem{acknowledgement}[theorem]{Acknowledgement} \newtheorem{algorithm}[theorem]{Algorithm} \newtheorem{axiom}[theorem]{Axiom} \newtheorem{case}[theorem]{Case} \newtheorem{claim}[theorem]{Claim} \newtheorem{conclusion}[theorem]{Conclusion} \newtheorem{condition}[theorem]{Condition} \newtheorem{conjecture}[theorem]{Conjecture} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{criterion}[theorem]{Criterion} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{exercise}[theorem]{Exercise} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{notation}[theorem]{Notation} \newtheorem{problem}[theorem]{Problem} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{remark}[theorem]{Remark} \newtheorem{solution}[theorem]{Solution} \newtheorem{summary}[theorem]{Summary} \newenvironment{proof}[1][Proof]{\noindent\textbf{#1.} }{\ \rule{0.5em}{0.5em}} \input{tcilatex} \geometry{left=0.6in,right=0.7in,top=0.7in,bottom=0.9in} \pagestyle{fancy} \lhead{\color{blue} \Large Lecture Notes} \lfoot{\small \copyright $\;$ copyright Hidegkuti, Powell, 2010} \rfoot{\small Last revised: October 18, 2010} \cfoot{} \chead{\LARGE Differentiation 2} \rhead{\large page \thepage} \textwidth 7.3in \textheight 9.3in \setlength{\headheight}{28pt} \begin{document} Differentiate each of the following functions.\bigskip %TCIMACRO{\TeXButton{2 col begin}{\begin{multicols}{2}}}% %BeginExpansion \begin{multicols}{2}% %EndExpansion \begin{enumerate} \item $f\left( x\right) =x^{5}\cos x$\medskip \item $f\left( x\right) =x^{3}\ln x$\medskip \item $f\left( x\right) =\sqrt[3]{x^{5}}\sin x$\medskip \item $f\left( x\right) =\left( 3x^{10}-5x^{6}\right) \ln x$\medskip \item $f\left( x\right) =x\ln x-x$\medskip \item $f\left( x\right) =\dfrac{\ln x}{x^{3}}$\medskip \item $f\left( x\right) =\ln \left( 8x^{10}\right) $\medskip \item $f\left( x\right) =\sin x\cdot \cos x$\medskip \item $f\left( x\right) =\ln x\cdot \sin x$\medskip \item $f\left( x\right) =\sin 2x$\medskip \item $f\left( x\right) =x^{10}\ln \left( x^{10}\right) $\medskip \item $f\left( x\right) =\cos x\left( x^{2}-4x+1\right) -\left( x^{3}-2\right) $\medskip \item $f\left( x\right) =\dfrac{\cos x}{x^{2}}$\medskip \item $f\left( x\right) =\dfrac{x^{3}-2x+1}{x^{2}}$\medskip \item $f\left( x\right) =\dfrac{\ln x}{x}$\medskip \item $f\left( x\right) =\dfrac{\ln \left( x^{3}\right) }{\sqrt{x^{3}}}$% \medskip \end{enumerate} %TCIMACRO{\TeXButton{2 col end}{\end{multicols}}}% %BeginExpansion \end{multicols}% %EndExpansion \pagebreak \begin{center} {\Large Answers}\bigskip \bigskip \end{center} 1.) \ $f^{\prime }\left( x\right) =-x^{5}\sin x+5x^{4}\cos x$ \ \ \ \ \ \ 2.) $\ f^{\prime }\left( x\right) =x^{2}+3x^{2}\ln x$\bigskip\ \ \ \ 3.) \ $f^{\prime }\left( x\right) =\left( \cos x\right) \sqrt[3]{x^{5}}+% \dfrac{5}{3}\sqrt[3]{x^{2}}\left( \sin x\right) $ \ \ \ 4.) \ \ $f^{\prime }\left( x\right) =\left( 30x^{9}-30x^{5}\right) \left( \ln x\right) +3x^{9}-5x^{5}$\bigskip 5.) \ \ $f^{\prime }\left( x\right) =\ln x$ \ \ \ \ \ \ \ \ \ 6.) \ $% f^{\prime }\left( x\right) =\dfrac{-3\ln x+1}{x^{4}}$\ \ \ \ \ \ \ 7.)\ \ $% f^{\prime }\left( x\right) =\dfrac{10}{x}$ \ \ \ \ \ 8.) \ $f^{\prime }\left( x\right) =\cos ^{2}x-\sin ^{2}x$\bigskip 9.) \ $f^{\prime }\left( x\right) =\ln x\cos x+\dfrac{1}{x}\sin x$ \ \ \ \ \ \ \ \ 10.) \ \ $f^{\prime }\left( x\right) =2\left( \cos ^{2}x-\sin ^{2}x\right) =2\cos 2x$\bigskip 11.) \ $f^{\prime }\left( x\right) =10x^{9}+100x^{9}\ln x$ \ \ \ \ \ \ \ 12.) \ $f^{\prime }\left( x\right) =$ $-\sin x\cdot \left( x^{2}-4x+1\right) +\left( \cos x\right) \left( 2x-4\right) -3x^{2}$\bigskip 13.)\ \ $f^{\prime }\left( x\right) =-\dfrac{1}{x^{2}}\sin x-\dfrac{2}{x^{3}}% \cos x$ \ \ \ \ \ \ \ \ \ \ \ \ 14.) \ $f^{\prime }\left( x\right) =1+\dfrac{% 2}{x^{2}}-\dfrac{2}{x^{3}}$ \ \ \ \ \ \ \ \ \ 15.) \ $f^{\prime }\left( x\right) =\dfrac{1-\ln x}{x^{2}}$\bigskip 16.) \ $f^{\prime }\left( x\right) =\dfrac{-\dfrac{9}{2}\ln x+3}{x^{5/2}}$ \end{document}