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\begin{document}
	
	\mbox{}
	\title{Problem Set 6: Differentiation Formulas}
	
	\maketitle
	
	
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%
	%\section{Introduction} \label{sec:intro}
	\noindent {\footnotesize Note: Most of the problems were taken from the textbook} \cite{jS15}.
	
%	\begin{problem}
%		Find the equation of a tangent line to the graph of the function $f(x) = x^2 + 4x + 3$ passing by the point $P = (0,-2)$.
%	\end{problem}

	\begin{problem}
		Differentiate the next functions.
		\begin{enumerate}[label=\alph*), topsep=8pt, itemsep=8pt, partopsep=1ex, parsep=1ex]
			\item $y = \sqrt[3]{2}(2+x)$;
			\item $y = \frac{\sqrt{x} + x}{x^2}$;
			\item $D(x) = \frac{1 + 16x^2}{(4x)^3}$.
		\end{enumerate}
	\end{problem}

	\begin{problem}
		Find the second derivative of the function $f(x) = \sqrt{x} + \sqrt[3]{x}$.
	\end{problem}

	\begin{problem}
		Find the points on the curve $y = 2x^3 + 3x^2 - 12x + 1$ where the tangent line is horizontal.
	\end{problem}

	\begin{problem}
		Find a second-degree polynomial $P$ such that $P(2) = 5$, $P'(2) = 3$, and $P''(2) = 2$.
	\end{problem}

\begin{problem}
	Find a cubic function $y = ax^3 + bx^2 + cx + d$ whose graph has horizontal tangent lines at the points $(-2,6)$ and $(2,0)$.
\end{problem}

	\begin{problem}
		Differentiate the next functions.
		\begin{enumerate}[label=\alph*), topsep=8pt, itemsep=8pt, partopsep=1ex, parsep=1ex]
			\item $f(x) = \frac{\sqrt{x}}{2+x}$;
			\item $f(x) = \frac{4+x}{xe^x}$;
			\item $f(x) = \frac{\sin x}{1 + \cos x}$;
			\item $f(x) = xe^x \cot x$.
		\end{enumerate}
	\end{problem}

	\begin{problem}
		Prove that $(\sec x)' = \sec x \tan x$.
	\end{problem}

	\begin{problem}
		For which values of $x$ does the graph of $f(x) = e^x \cos x$ have horizontal tangent line?
	\end{problem}

	\begin{problem}
		Find the following limits:
		\[
			\lim_{x \to 0} \frac{\sin 3x \sin 5x \sin 7x}{x^3} \ \ \text{ and } \ \ \lim_{x \to 1} \frac{\sin(x-1)}{x^2 + x - 2}.
		\]
	\end{problem}

	\begin{problem}
		Find $\frac{d^{35}}{dx^{35}}(x \sin x)$.
	\end{problem}
	
	\vspace{20pt}
	
	\begin{thebibliography}{20}
		
		\bibitem{jS15} J.~Stewart: \emph{Single Variable Calculus} 8th Edition, Cengage Learning, Boston 2015.
		
	\end{thebibliography}
	
\end{document}