\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}{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=0.6in,top=0.6in,bottom=0.7in} \pagestyle{fancy} \lhead{\color{blue} \Large Lecture Notes} \chead{\color{black} \LARGE Differentiation 5} \rhead{\large page \ \thepage} \cfoot{} \lfoot{\small \copyright $\;$ copyright Hidegkuti, Powell, 2009} \rfoot{\small Last revised: March 20, 2012} \textwidth 7.5in \textheight 9.6in \setlength{\headheight}{28pt} \setlength{\parindent}{0in} \begin{document} \begin{center} {\Large Practice Problems\bigskip }\bigskip \end{center} Differentiate each of the following functions. \ \ Assume that $a$ is a constant. \ \ Please note that the inverse function for $\sin x,$ \ (sometimes denoted by $\sin ^{-1}x)$ \ is denoted by $\arcsin x$ here.\bigskip \bigskip %TCIMACRO{\TeXButton{3 col begin}{\begin{multicols}{3}}}% %BeginExpansion \begin{multicols}{3}% %EndExpansion \begin{enumerate} \item $f\left( x\right) =e^{3-x}$\medskip \item $f\left( x\right) =\arctan x$\medskip \item $f\left( x\right) =\sqrt{a^{2}-\sin ^{2}x}$\medskip \item $f\left( x\right) =\arcsin 3x$ \item $f\left( x\right) =2^{5x^{2}+1}$\medskip \item $f\left( x\right) =\ln \left( 5x^{2}-8x+3\right) $\medskip \item $f\left( x\right) =-\dfrac{\sin x\cos x}{2}+\dfrac{x}{2}$\medskip \item $f\left( x\right) =e^{-x^{2}}$\medskip \item $f\left( x\right) =\cos x\cdot 10^{\sin x}$\medskip \item $f\left( x\right) =-\dfrac{1}{4}e^{-2x}-\dfrac{1}{2}xe^{-2x}$\medskip \item $f\left( x\right) =\dfrac{1}{2}\left( \arcsin x\right) ^{2}+e^{5}$% \medskip \item $f\left( x\right) =10\sin 2x\cos 2x$\medskip \item $f\left( x\right) =\dfrac{1}{2}x^{2}\ln x-\dfrac{1}{4}x^{2}$\medskip \item $f\left( x\right) =\arctan \left( 10x\right) $\medskip \item $f\left( x\right) =\ln \left( \tan x\right) $\medskip \item $f\left( x\right) =\dfrac{x}{x+1}$\medskip \item $f\left( x\right) =\arctan \left( \dfrac{x}{x+1}\right) $\medskip \item $f\left( x\right) =\arcsin \left( x^{2}-1\right) $\medskip \item $f\left( x\right) =\log _{3}\left( \sec x\right) $\medskip \item \ $f\left( x\right) =\cos ^{-1}\left( x^{6}\right) $\medskip \item \ $f\left( x\right) =2^{\sin \left( -5x\right) }$\medskip \item $f\left( x\right) =e^{\sin x+\cos x}$\medskip \item $f\left( x\right) =\sin \left( \ln \left( x^{3}\right) \right) $% \medskip \end{enumerate} %TCIMACRO{\TeXButton{3 col end}{\end{multicols}}}% %BeginExpansion \end{multicols}% %EndExpansion \pagebreak \begin{center} {\Large Practice Problems - Answers \bigskip }\bigskip \bigskip \end{center} %TCIMACRO{\TeXButton{2 col begin}{\begin{multicols}{2}}}% %BeginExpansion \begin{multicols}{2}% %EndExpansion \begin{enumerate} \item $f^{\prime }\left( x\right) =-e^{3-x}$\medskip \item $f^{\prime }\left( x\right) =\dfrac{1}{x^{2}+1}$\medskip \item $f^{\prime }\left( x\right) =-\dfrac{\cos x\sin x}{\sqrt{a^{2}-\sin ^{2}x}}$\medskip \item $f^{\prime }\left( x\right) =\dfrac{3}{\sqrt{1-9x^{2}}}$\medskip \item $f^{\prime }\left( x\right) =10x\left( \ln 2\right) 2^{5x^{2}+1}$% \medskip \item $f^{\prime }\left( x\right) =\dfrac{10x-8}{5x^{2}-8x+3}$\medskip \item $f^{\prime }\left( x\right) =\sin ^{2}x$\medskip \item $f^{\prime }\left( x\right) =-2xe^{-x^{2}}$\medskip \item $f^{\prime }\left( x\right) =10^{\sin x}\left[ -\sin x+\left( \ln 10\right) \cos ^{2}x\right] $ \item $f^{\prime }\left( x\right) =xe^{-2x}$\medskip \item $f^{\prime }\left( x\right) =\dfrac{\arcsin x}{\sqrt{1-x^{2}}}$% \medskip \item $f^{\prime }\left( x\right) =20\cos ^{2}2x-20\sin ^{2}2x=20\cos 4x$% \medskip \item $f^{\prime }\left( x\right) =x\ln x$\medskip \item $f^{\prime }\left( x\right) =\dfrac{10}{100x^{2}+1}$\medskip \item $f^{\prime }\left( x\right) =\dfrac{\tan ^{2}x+1}{\tan x}=\tan x+\cot x $\medskip \item $f^{\prime }\left( x\right) =\dfrac{1}{\left( x+1\right) ^{2}}$ \item $f^{\prime }\left( x\right) =\dfrac{1}{2x^{2}+2x+1}$ \item $f^{\prime }\left( x\right) =\dfrac{2x}{\sqrt{1-\left( x^{2}-1\right) ^{2}}}$ \item $f^{\prime }\left( x\right) =\dfrac{\tan x}{\ln 3}$ \item \ $f^{\prime }\left( x\right) =-\dfrac{6x^{5}}{\sqrt{1-x^{12}}}$ \item $f^{\prime }\left( x\right) =-5\left( \ln 2\right) \dfrac{\cos 5x}{% 2^{\sin 5x}}$ \item $f^{\prime }\left( x\right) =e^{\cos x+\sin x}\left( \cos x-\sin x\right) $ \item $f^{\prime }\left( x\right) =\allowbreak \dfrac{3}{x}\cos \left( \ln x^{3}\right) $ \end{enumerate} %TCIMACRO{\TeXButton{3 col end}{\end{multicols}}}% %BeginExpansion \end{multicols}% %EndExpansion \bigskip \end{document}