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\centerline{Mathematics 136 -- Advanced Placement Calculus}
\centerline{Discussion 2 -- Review of Differentiation Formulas}
\centerline{September 16, 2009}
\vskip 10pt
\noindent
{\it Background}
\vskip 10pt
Over the last few days we have been thinking about the definition
of the derivative of $f(x)$ at $x$:
\[
f'(x) = \lim_{h\to 0} \frac{f(x+h) - f(x)}{h}.
\]
Of course, if we had to use the 
definition every time we wanted to compute a derivative, it
would be very awkward and time consuming.  From your high school
calculus course, you should recall a number of methods for 
computing derivatives of functions defined by formulas.  We
want to review those today and work some examples.  
{\it The goal is to finish work on this and hand it in by 
the end of today's class meeting.}
\vskip 10pt
\begin{enumerate}
\item[A.]  {\it The general rules}.  Say you know that the 
derivatives of $f(x)$ and $g(x)$ exist at $x$.  
\begin{enumerate}
\item[1.]  Say {\it in words, and with a formula} how you would find the derivative of the function 
$cf(x) + g(x)$ ($c$ constant) at $x$.
\item[2.]  Say {\it in words, and with a formula} how you would find the derivative of the product 
function $f(x)g(x)$ at $x$.
\item[3.]  Say {\it in words, and with a formula} how you would find the derivative of the 
quotient function $\frac{f(x)}{g(x)}$ at $x$.
\item[4.]  Say {\it in words, and with a formula} how you would find the derivative of the 
composite function $f(g(x))$ at $x$.
\end{enumerate}
\item[B.]  Use the rules from part A, and derivatives for known functions like
powers $x^n$, exponentials $e^x$, $\ln(x)$, $\sin(x)$, $\cos(x)$, and so on to find derivatives for
each of the following.  In each case, show all work, write the calculations
carefully as equations of the form $f'(x) = \cdots$, and say which of the rules from part A 
you are using.
\begin{enumerate}
\item[1.]  $\displaystyle{f(x) = 6x^3 + 7 x^{\frac{1}{2}} - 3 x^{-\frac{5}{3}} + \pi^e}$.
\item[2.]  $\displaystyle{g(x) = e^x - \sqrt{x} + \frac{1}{x^7}}$.
\item[3.]  $\displaystyle{h(x) = e^x\cos(x)}$.
\item[4.]  $\displaystyle{k(x) = \frac{x^2 + 1}{x^4 + 2}}$.
\item[5.]  $\displaystyle{\ell(x) = x\ln(5x^2 + 1)}$.
\item[6.]  $\displaystyle{m(x) = \sqrt{e^x + 1}}$. 
\item[7.]  $\displaystyle{n(x) = \cos(\sin(x^3))}$.
\item[8.]  $\displaystyle{p(x) = \left(x + \frac{1}{x}\right)^4}$.
\end{enumerate}
\end{enumerate}
\end{document}
