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%  Differentiation Rules: Trigonometric Functions                  %
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\problemgrouptitle{Differentiation Rules: Trigonometric Functions}

\item Find the derivative $f'(x)$ of $f(x)=\sin 4x$ using the definition of the derivative.
\answer{Use the sine addition identity followed by the fundamental limits involving sine and cosine to get $f'(x)=4\cos 4x$~.}

\begin{problemblock}[2]{Differentiate the following functions involving trigonmetric functions.} 

\item $f(x)=x^2\cos x$
\answer{$f'(x)=2x\cos x-x^2\sin x$}

\item $f(t)=\displaystyle\frac{t^3}{\sin t+\tan t}$
\answer{$f'(t)=\displaystyle\frac{3t^2(\sin t+\tan t)-t^3\left(\cos t+\sec^2t\right)}{(\sin t+\tan t)^2}$}

\item $H(\theta)=\csc\theta\cot\theta$ ; Also find $\displaystyle \left.\frac{dH}{d\theta}\right|_{\theta=\pi/3}$~.
\answer{$\displaystyle \frac{dH}{d\theta}=-\csc\theta\cot^2\theta-\csc^3\theta$ ; $\displaystyle H'(\pi/3)=-\frac{10}{3\sqrt{3}}$}

\item $f(x)=(\sin x+\cos x)(\sec x-\cot x)$
\answer{$f'(x)=(\cos x-\sin x)(\sec x-\cot x)+(\sin x+\cos x)(\sec x\tan x+\csc^2x)$}

\end{problemblock}

\item Calculate $\displaystyle\frac{df}{d\theta}$ for the function $f(\theta)=\sin^2\theta+\cos^2\theta$
\begin{enumerate}
\item Directly by using the rules of differentiation.
\item By first simplifying $f$ with a trigonometric identity and then differentiating.
\end{enumerate}
\answer{$\displaystyle \frac{df}{d\theta}=0$}

\begin{problemblock}{For each curve and point $P$,\newline\begin{inparaenum}\item Confirm the point $P$ lies on the curve. \item Find the equation of the tangent line at $P$\,.\end{inparaenum}}

\item $y=x\tan x$ at point $\displaystyle P\left(\frac{\pi}{4},\frac{\pi}{4}\right)$
\answer{\begin{enumerate}
\item Since $\tan(\pi/4)=1$, $x=\pi/4$ and $y=\pi/4$ indeed satisfy the equation.
\item Point-slope form: $y=(1+\pi/2)(x-\pi/4)+\pi/4$ , Slope-intercept form: $y=(1+\pi/2)x-\pi^2/8$
\end{enumerate}}

\item $y=5x+3\sin(2x)-2\cos(3x)$ at point $P(0,-2)$.
\answer{\begin{enumerate}
\item $-2=5(0)+3(0)-2(1)$
\item $y=11x-2$
\end{enumerate}
}

\end{problemblock}
