TeX code compiled with \documentclass{beamer} using the Amsterdam theme.\\ \\ 

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\begin{document}

\begin{frame}
    \large You own a company producing iSquids, (the latest portable electronic craze). 
    Your big production limitation is a scarcity of Chip 187, produced by outside manufacturers. 
    \vskip 5pt
    
    If $f(x)$ is the profit your company will make if it gets $x$ Chip 187's and $g(x)$ is a 
    function giving the number of Chip 187's you can obtain for $x$ dollars, 
    which of the following is of interest to you?
    \vskip 8pt

    \begin{columns}
    \begin{column}{0.5\textwidth}
    \begin{itemize}
    \item[\bf (a)] $f\circ g$
    \item[\bf (b)] $g\circ f$
    \end{itemize}
    \end{column}

    \begin{column}{0.5\textwidth}
    \begin{itemize}
    \item[\bf (c)] both
    \item[\bf (d)] neither
    \end{itemize}
    \end{column}
    \end{columns}
\end{frame}

\begin{frame}
    \Large 
    $$ f(x)=x+\frac{1}{x} \hskip 30pt g(x)=\dfrac{x+8}{x+2} \hskip 30pt h(x)=\sqrt{x} $$

    Express each function as an equation. \\ What is the domain of each function?
    \vskip 10pt

    \begin{columns}
    \begin{column}{0.5\textwidth}
    $(f\circ g)(x)$
    \vskip 30pt

    $g(f(x))$
    \vskip 30pt

    $(g\circ g)(x)$
    \end{column}

    \begin{column}{0.5\textwidth}
    $(h\circ f)(x)$
    \vskip 30pt

    $(g\circ h)(x)$
    \vskip 30pt

    $h(h(x))$
    \end{column}
    \end{columns}
    \vskip 20pt

\end{frame}


\begin{frame}
    \large For each of the following functions, first express it as a composition of 2 functions. 
    Then find the derivatives.
    \vskip 15pt

    \begin{columns}
    \begin{column}{0.5\textwidth}
    \begin{enumerate}
    \item[\bf a)] $F(x)=\sqrt[3]{1+5x}$
    \vskip 30pt

    \item[\bf b)] $G(x)=(x^4+9x^2+3)^8$
    \vskip 30pt

    \item[\bf c)] $F(t)=\sqrt[9]{1+\tan(t)}$
    \end{enumerate}
    \end{column}

    \begin{column}{0.5\textwidth}
    \begin{enumerate}
    \item[\bf d)] $H(x)=\cos(3^7+x^7)$
    \vskip 30pt

    \item[\bf e)] $G(x)=\left(\dfrac{x^2+8}{x^2-8}\right)^3$
    \vskip 30pt

    \item[\bf f)] $S(z)=\sqrt{\dfrac{z-7}{z+7}}$
    \end{enumerate}
    \end{column}
    \end{columns}
\end{frame}


\begin{frame}
    \large Find the derivatives.
    \vskip 15pt

    \begin{columns}
    \begin{column}{0.45\textwidth}
    \begin{enumerate}
    \item[\bf a)] $y=\dfrac{r}{\sqrt{r^2+3}}$
    \vskip 20pt

    \item[\bf b)] $y=x\sin\left(\dfrac{7}{x}\right)$
    \vskip 20pt

    \item[\bf c)] $f(t)=\sqrt{\dfrac{t}{t^2+1}}$
    \vskip 20pt

    \item[\bf d)] $g(y)=\dfrac{(y-2)^6}{(y^2+4y)^9}$
    \end{enumerate}
    \end{column}

    \begin{column}{0.55\textwidth}
    \begin{enumerate}
    \item[\bf e)] $y=\sin(\tan(8x))$
    \vskip 20pt

    \item[\bf f)] $y=\cos(\cos(\cos(x)))$
    \vskip 20pt

    \item[\bf g)] $y=(1+\sec(3\pi x+4\pi))^5$
    \vskip 20pt

    \item[\bf h)] $y=\sqrt{11x+ \sqrt{11x+ \sqrt{11x}}}$
    \vskip 20pt

    \item[\bf i)] $y = [x + (x + \sin(2 x))^6]^7$
    \end{enumerate}
    \end{column}
    \end{columns}
\end{frame}

\begin{frame}
    \large If  $h(x) = \sqrt{7 + 6f(x)}$,
     where
    \begin{center}
    $f(4) = 7$ and $f '(4) = 2$,
    \end{center}
     find $h'(4)$.
    \vskip 70pt

    Find the first and second derivatives of $y=\sin\left(x^2\right)$.
\end{frame}


\begin{frame}
    \large If $f$ and $g$ are both differentiable and $h=f\circ g$, $h^{\prime}(2)$ equals
    \vskip 20pt

    \begin{enumerate}
    \item $f^{\prime}(2)\circ g^{\prime}(2)$
    \item $f^{\prime}(2)g^{\prime}(2)$
    \item $f^{\prime}(g(2)) g^{\prime}(2)$
    \item $f^{\prime}(g(x)) g^{\prime}(2)$
    \end{enumerate}
\end{frame}

\end{document} 
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