\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=0.4in,right=0.5in,top=0.5in,bottom=0.4in} \pagestyle{fancy} \lhead{\color{blue} \large Practice} \chead{\color{black} \Large Differentiation 1} \rhead{\small page \ \thepage} \cfoot{} \lfoot{\footnotesize \copyright $\;$ Hidegkuti, Powell, 2011} \rfoot{\footnotesize Last revised: October 3, 2017} \textwidth 7.7in \textheight 9.7in \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 $a,b,$ and $c$ are constants.\bigskip %TCIMACRO{\TeXButton{2 col begin}{\begin{multicols}{2}}}% %BeginExpansion \begin{multicols}{2}% %EndExpansion \begin{enumerate} \item $f\left( x\right) =3x^{2}-5x+8$ \item $f\left( x\right) =ax^{2}+bx+c$ \item $f\left( x\right) =3\sin x-2x+1$ \item $f\left( x\right) =\dfrac{1}{x^{3}}$ \item \ $f\left( x\right) =\sqrt[5]{x^{2}}$ \item \ $f\left( x\right) =-\left( x+1\right) ^{2}$ \item $f\left( x\right) =-\cos x$ \item $f\left( x\right) =x^{2}-\dfrac{1}{x^{2}}$ \item $f\left( x\right) =-\dfrac{2}{3}\left( x^{7}+2x-15\right) $ \item $f\left( x\right) =mx+b$ \item $f\left( x\right) =\sqrt{x^{7}}-\dfrac{1}{x^{7}}-e^{7}$ \item $f\left( x\right) =\sin ^{2}x+\cos ^{2}x$ \item $f\left( x\right) =\dfrac{\left( x+5\right) ^{3}+\left( x-5\right) ^{3}% }{2}$ \item $f\left( x\right) =3\pi ^{5}-3\pi +5$ \item $f\left( x\right) =\dfrac{\left( x+5\right) ^{3}-\left( x-5\right) ^{3}% }{2}$ \item $f\left( x\right) =\sqrt[7]{x^{3}}+\sqrt[3]{x^{7}}$ \item $f\left( x\right) =x^{10}+\sqrt[10]{x}+\dfrac{1}{\sqrt[10]{x}}+\dfrac{1% }{x^{10}}$ \item $f\left( x\right) =3x^{8}-4x^{5}+x^{2}-6$ \item $f\left( x\right) =-\dfrac{1}{x^{2}}\left( 2x^{3}+4\right) $ \item $f\left( x\right) =x^{a}+\sqrt[b]{x}$ \item $f\left( x\right) =x+1-\dfrac{1}{x}$ \item $f\left( x\right) =5x^{4}-\pi ^{2}+3x-\dfrac{1}{e}$ \item $f\left( x\right) =\dfrac{3}{x}-x+\left( x-e\right) ^{2}$ \item $f\left( x\right) =\pi x^{2}-e^{3}-\dfrac{5x}{\sqrt[3]{x^{2}}}$ \end{enumerate} %TCIMACRO{\TeXButton{2 col end}{\end{multicols}}}% %BeginExpansion \end{multicols}% %EndExpansion \bigskip \bigskip \begin{enumerate} \item[25.] Find a function $f\left( x\right) $ so that\bigskip a) \ $f^{\prime }\left( x\right) =6x$ \ \ \ \ \ \ \ \ b) \ $f^{\prime }\left( x\right) =x^{2}+x$ \ \ \ \ \ \ \ \ c) \ $f^{\prime }\left( x\right) =% \dfrac{1}{x^{5}}$\pagebreak \end{enumerate} \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 $f^{\prime }\left( x\right) =6x-5$ \item \ $f^{\prime }\left( x\right) =2ax+b$ \item $f^{\prime }\left( x\right) =3\cos x-2$ \item $f^{\prime }\left( x\right) =-\dfrac{3}{x^{4}}$ \item $f^{\prime }\left( x\right) =\dfrac{2}{5}x^{-3/5}=\dfrac{2}{5\sqrt[5]{% x^{3}}}=\dfrac{2}{5x}\sqrt[5]{x^{2}}$ \item $f^{\prime }\left( x\right) =-2x-2$ \item $f^{\prime }\left( x\right) =\sin x$ \item $f^{\prime }\left( x\right) =2x+\dfrac{2}{x^{3}}$ \item $f^{\prime }\left( x\right) =-\dfrac{14}{3}x^{6}-\dfrac{4}{3}$ \item $f^{\prime }\left( x\right) =m$ \item $f^{\prime }\left( x\right) =\dfrac{7}{2}x^{5/2}+7x^{-8}$ \item $f^{\prime }\left( x\right) =0$ \item $f^{\prime }\left( x\right) =3x^{2}+75$ \item $f^{\prime }\left( x\right) =0$ \item $f^{\prime }\left( x\right) =30x$ \item $f^{\prime }\left( x\right) =\dfrac{3}{7}x^{-4/7}+\dfrac{7}{3}x^{4/3}$ \item $f^{\prime }\left( x\right) =10x^{9}+\dfrac{1}{10\sqrt[10]{x^{9}}}-% \dfrac{1}{10x\sqrt[10]{x}}-\dfrac{10}{x^{11}}$ \item $f^{\prime }\left( x\right) =24x^{7}-20x^{4}+2x$ \item $f^{\prime }\left( x\right) =-2+\dfrac{8}{x^{3}}$ \item $f^{\prime }\left( x\right) =ax^{a-1}+\dfrac{1}{b}\dfrac{\sqrt[b]{x}}{x% }$ \item $f^{\prime }\left( x\right) =1+\dfrac{1}{x^{2}}$ \item $f^{\prime }\left( x\right) =20x^{3}+3$ \item $f^{\prime }\left( x\right) =2x-2e-1-\dfrac{3}{x^{2}}$ \item $f^{\prime }\left( x\right) =2\pi x-\dfrac{5}{3\sqrt[3]{x^{2}}}$ \end{enumerate} %TCIMACRO{\TeXButton{2 col end}{\end{multicols}}}% %BeginExpansion \end{multicols}% %EndExpansion {\Large \bigskip }\bigskip \begin{enumerate} \item[25.] a) \ \ $f\left( x\right) =3x^{2}$ \ \ \ \ \ b) \ \ $f\left( x\right) =\dfrac{1}{3}x^{3}+\dfrac{1}{2}x^{2}$ \ \ \ \ \ \ c) \ $f\left( x\right) =-\dfrac{1}{4x^{4}}$ \end{enumerate} \bigskip \vspace{2.6in} {\small %TCIMACRO{\TeXButton{\small}{\small}}% %BeginExpansion \small% %EndExpansion } \href{https://teaching.martahidegkuti.com/shared/lnotes/lecturenotes.html}{% For more documents like this, visit our page at\ https://teaching.martahidegkuti.com and click on Lecture Notes. \ E-mail questions or comments to mhidegkuti@ccc.edu.} \end{document}