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\begin{document}
	
	\mbox{}
	\title{Problem Set 7: Chain Rule and Implicit Differentiation}
	
	\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 $y = \sin(\tan 2x)$;
			\item $y = \frac{\cos \pi x}{ \sin \pi x + \cos \pi x}$;
			\item $f(x) = \sqrt{x + \sqrt{x + \sqrt{x}}}$.
		\end{enumerate}
	\end{problem}

	\begin{problem}
		Find the points on the graph of the function $f(x) = 2 \sin x + \sin^2 x$ at which the tangent line is horizontal.
	\end{problem}

	\begin{problem}
	If $F(x) = f(3f(4f(x)))$, where $f(0) = 0$ and $f'(0) = 2$, find $F'(0)$.
	\end{problem}

	\begin{problem}
		If $f(xf(xf(x)))$, where $f(1) = 2$, $f(2) = 3$, $f'(1) = 4$, $f'(2) = 5$, and $f'(3) = 6$, find $F'(1)$.
	\end{problem}

	\begin{problem}
		Find $dy/dx$ by implicit differentiation.
		\begin{enumerate}[label=\alph*), topsep=8pt, itemsep=8pt, partopsep=1ex, parsep=1ex]
			\item $\tan(x/y) = x+y$;
			\item $\sqrt{xy} = 1 + x^2y$;
			\item $\tan(x-y) = \frac{y}{1 + x^2}$.
		\end{enumerate}
	\end{problem}

	\begin{problem}
		If $xy + y^3 = 1$, find the value of $y''$ at the point where $x=0$.
	\end{problem}

	\begin{problem}
		Find an equation of the tangent line of the hyperbola
		\[
			\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1
		\]
		at the point $(x_0, y_0)$.
	\end{problem}

	\begin{problem}
		Find $y'$ if $x^y = y^x$.
	\end{problem}
	
	\begin{problem}
		 Find a formula for $f^{(n)}(x)$ if $f(x) = \ln(x-1)$.
	\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}