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\problemgrouptitle{Differentiation Review}

\begin{problemblock}{For each function calculate $f'(x)$ using the definition of the derivative.}

\item $\displaystyle f(x)=x^3+2$
\answer{$\displaystyle f'(x)=3x^2$}

\item $\displaystyle f(x)=\frac{4x-3}{x+2}$
\answer{$\displaystyle f'(x)=\frac{11}{(x+2)^2}$}

\item $\displaystyle f(x)=\sqrt{2x+1}$
\answer{$\displaystyle f'(x)=\frac{1}{\sqrt{2x+1}}$}
\end{problemblock}

\begin{problemblock}{Differentiate the functions.}

\item $\displaystyle y=3x^4+\sqrt[3]{2x}-\frac{5}{\sqrt{x}}+\pi$
\answer{$\displaystyle y'=12x^3+\frac{\sqrt[3]{2}}{3}x^{-2/3}+\frac{5}{2}x^{-3/2}$}

\item $\displaystyle g(x)=\left(\sqrt{2x}-4x+3\right) \left(3x+\sin x\right)$
\answer{$\displaystyle g'(x)=\left(\frac{1}{\sqrt{2x}}-4\right) \left(3x+\sin x\right) +\left(\sqrt{2x}-4x+3\right)\left(3+\cos x\right)$}

\item $\displaystyle h(y)=\frac{\sqrt{y+5}}{3y+2}$
\answer{$\displaystyle h'(y)=\frac{-3y-10}{2\sqrt{y+5}\left(3y+2\right)^2}$}

\item $\displaystyle f(\theta )=\cos^2\theta+4\cos\!\left(\theta^2\right)$
\answer{$\displaystyle f'(\theta)=-2\cos\theta\sin\theta-8\theta\sin\!\left(\theta^2\right)$}

\item $\displaystyle g(x)=\sec\!\left(x^3+4\right)\cos(2x)$
\answer{$\displaystyle g'(x)=3x^2\sec\!\left(x^3+4\right)\tan\!\left(x^3+4\right)\cos(2x)-2\sec\!\left(x^3+4\right)\sin(2x)$}

\item $\displaystyle f(x)=\sqrt[5]{\frac{x^3-4x+10}{4x^2+5}}$
\answer{$\displaystyle f'(x)=\frac{1}{5}\left(\frac{x^3-4x+10}{4x^2+5}\right)^{-4/5}\frac{4x^4+31x^2-80x-20}{\left(4x^2+5\right)^2}$}

\item $\displaystyle x^4y^3+4y^2=xy+\sin y$
\answer{$\displaystyle \frac{dy}{dx}=\frac{y-4x^3y^3}{3x^4y^2+8y-x-\cos y}$}

\end{problemblock}

\item Find the equation of the tangent line to the curve $\displaystyle y=3\sin x-2\cos (3x)$ at the point $P(\frac{\pi}{2},3)$.
\answer{Point-slope form: $y=-6(x-\pi/2)+3$, Slope-intercept form: $y=-6x+3\pi+3$}

\item Find the value(s) of $x$ for which the curve $\displaystyle y=\frac{x+1}{x^2+3}$ has a horizontal tangent line. 
\answer{$x=1$, $x=-3$}

\item Find the value(s) of $\theta $ for which the curve $\displaystyle f(\theta )=\cos(2\theta)-2\cos\theta$ has a horizontal tangent line.
\answer{$\displaystyle \theta =\frac{\pi}{3}+2n\pi$, $\displaystyle \theta =\frac{5\pi}{3}+2n\pi$, $\displaystyle \theta=n\pi$  ($n$ an integer)}

\item The height $h$ and radius $r$ of a circular cone are increasing at the rate of $3$~cm/sec. How fast is the volume of the cone increasing when $h=8$~cm and $r=3$~cm?
\answer{$57\pi$~cm$^3$/sec}

\item A right triangle has a constant height of $30$~cm. If the base of the right triangle is increasing at a rate of $6$~cm/sec, how fast is the angle between the hypotenuse and the base changing when the base is $30$~cm?
\answer{$\displaystyle -\frac{1}{10}$~rad/sec}

\item If the area of an equilateral triangle is increasing at a rate of 5~cm$^2/\sec$, find the rate at which the length of a side is changing when the area of the triangle is 100 cm$^3$.
\answer{$\displaystyle \frac{1}{2(3)^{1/4}}$~cm/sec}
