13-lesson.tex

\section*{Derivative Review. Trig Functions}

Basic Formulas
\begin{equation*}
  \begin{aligned}
    \frac{d}{dx} \sin x &= \cos x \\
    \frac{d}{dx} \cos x &= \sin x \\
    \frac{d}{dx} \tan x &= \frac{\sin x}{\cos x} \\
    \noalign{Use quotient rule}
    &= \frac{\cos x \cos x - \sin x (-\sin x)}{\cos^2 x} \\
    &= \frac{\cos^2 x + \sin^2 x}{\cos^2 x} \\
    \noalign{Use pythogorean trig identity $\sin^2 + \cos^2 =1$}
    &= \frac{1}{\cos^2 x}=\sec^2 x\\
  \end{aligned}
\end{equation*}
\begin{questions}
\question
Calculate the derivative of $\sin x \cos x$ two ways: use product rule and use the trig identity $\sin (2x) = 2\sin x\cos x$. Do you get the same answer?
\begin{solution}[1.5in]
\end{solution}
\question
Calculate the derviative of $\csc x$ hint $\csc = \frac{1}{\sin}$
\begin{solution}[1.5in]
\end{solution}
\question
Calculate the derviative of $\cos (x^2+1)$
\begin{solution}[1.5in]
\end{solution}
\end{questions}