%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                                                                  %
%  Differentiation Rules: Power, Sums, Constant Multiple           %
%                                                                  %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\problemgrouptitle{Differentiation Rules: Power, Sums, Constant Multiple}

\begin{problemblock}[2]{Differentiate the following functions involving powers, sums and constant multiplication.  (Any value that is not the function variable should be considered a constant.)}
\addtoanswerkey{\begin{multicols}{2}}
\item $f(x)=\sqrt{x}-x^{12}$
\answer{$\displaystyle f'(x)=\frac{1}{2}x^{-\frac{1}{2}}-12x^{11}=\frac{1}{2\sqrt{x}}-12x^{11}$}

\item $g(x)=\displaystyle\frac{1}{\sqrt{x^5}}$
\answer{$\displaystyle g'(x)=-\frac{5}{2}x^{-\frac{7}{2}}=\frac{-5}{2\sqrt{x^7}}$}

\item $y=\displaystyle\frac{4x^3}{5}$
\answer{$\displaystyle\frac{dy}{dx}=\frac{12}{5}x^2$}

\item $f(u)=u^{-4}+u^4$ ; Also find $f'(1)$ .
\answer{$f'(u)=-4u^{-5}+4u^3$ ; $f'(1)=0$}

\item $\displaystyle f(x)=(x^3+x)^2$ (Hint: Expand first.)
\answer{$f'(x)=6x^5+8x^3+2x$}

\item $\displaystyle f(x)=\sqrt{3x}+\sqrt[5]{\frac{x}{3}}$\\ (Hint: $\displaystyle\sqrt[n]{xy}=(\sqrt[n]{x})(\sqrt[n]{y})$ and $\displaystyle\sqrt[n]{\frac{x}{y}}=\frac{\sqrt[n]{x}}{\sqrt[n]{y}}$\,.)
\answer{$\displaystyle f'(x)=\sqrt{3}\cdot\frac{1}{2}x^{-\frac{1}{2}}+\frac{1}{\sqrt[5]{3}}\cdot\frac{1}{5}x^{-\frac{4}{5}}=\frac{\sqrt{3}}{2\sqrt{x}}+\frac{1}{5\sqrt[5]{3x^4}}$}

\item $\displaystyle f(x)=\sin(\pi/15)x^{2a}$
\answer{$\displaystyle f'(x)=2a\sin(\pi/15)x^{2a-1}$}

\item $\displaystyle s(t)=-\frac{g}{2}t^2+v_{\scriptscriptstyle 0}t+s_{\scriptscriptstyle 0}$
\answer{$\displaystyle \frac{ds}{dt}=-gt+v_{\scriptscriptstyle 0}$}

\addtoanswerkey{\end{multicols}}

\end{problemblock}

\item Calculate the instantaneous rates of change given in problems \ref{itm:estimateslope}, \ref{itm:estimatevelocity}, and \ref{itm:estimatevolumechange} of \lab{Section}\textbook{Exercise}~\ref{DIFFERENTIATION/average-and-instantaneous-rate-of-change} directly using the derivative.
\answer{\begin{description}
\item[\normalfont\ref*{itm:estimateslope}:]$\displaystyle\left.\frac{dy}{dx}\right|_{x=2}=\left[3x^2+1\right|_{x=2}=13$
\item[\normalfont\ref*{itm:estimatevelocity}:]$v(2)=\displaystyle\left.\frac{ds}{dt}\right|_{t=2}=\left[3t^2+4t\right|_{t=2}=20$~cm/s
\item[\normalfont\ref*{itm:estimatevolumechange}:]$\displaystyle\left.\frac{dV}{dP}\right|_{P=3}=\left.-(22.4)\cdot P^{-2}\right|_{P=3}=-2.49$ L/atm
\end{description}}

\item Find the equation of the tangent line to the curve $y=x+\sqrt{x}$ at the point $P(1,2)$.
\answer{$\displaystyle y=\frac{3}{2}x+\frac{1}{2}$}

\item Find the value(s) of $x$ for which the curve $y=2x^3-4x^2+5$ has a horizontal tangent line.
\answer{$\displaystyle x=0,\ \frac{4}{3}$}
