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\centerline{MATH 135 -- Calculus 1}
\centerline{The Derivative of a Function}
\centerline{October 9 and 11, 2019}
\vskip 10pt
\noindent
{\it Initial Background}
\vskip 10pt
We are now ready to begin Chapter 3 in our textbook.
In the videos for today's class, we introduced the \emph{derivative} of a function 
$f$ at $x = a$ in the domain of $f$:
\begin{equation}
\label{der}
f'(a) = \lim_{h\to 0} \frac{f(a + h) - f(a)}{h} = \lim_{x \to a} \frac{f(x) - f(a)}{x - a},
\end{equation}
$f'(a)$ will give the slope of the tangent line 
to the graph $y = f(x)$ at the point $(a,f(a))$.  If $x$ represents time, and $f(x)$ is a position,
then $f'(a)$ would be the instantaneous velocity.  

All the techniques we learned in Chapter 2 for 
computing indeterminate form limits were, in fact, \emph{set up to compute the limits giving}
$f'(a)$(!)  To start, let's practice (and review) some of those techniques!

\vskip 10pt
\noindent
{\it Questions}
\vskip 10pt
\noindent
\begin{enumerate}
\item[(1)]  \emph{Using the limit definition} from \eqref{der} above, 
compute $f'(a)$ for $f(x) = x^3 + 2x + 1$--the derivative at
a general $x = a$ for $f(x) = x^3 + 2x + 1$.  Use your result to 
find the equation of the tangent line to the graph $y = x^3 + 2x + 1$ at the point $(1,4)$.  
\item[(2)]  Compute $f'(a)$ for $f(x) = \sqrt{x + 1}$--the derivative at
a general $x = a$ for $f(x) = \sqrt{x + 1}$.  Here there is a restriction on which $a$
``work.''  What is that restriction?  Does this make sense, thinking of the graph $y = \sqrt{x + 1}$?
(Note:  this is part of the parabola with equation $x = y^2 - 1$.)  Use your result to 
find the equation of the tangent line to the graph $y = \sqrt{x + 1}$ at the point $(3,2)$. 
\end{enumerate}
\noindent
{\it Further Background}
\vskip 10pt
In today's videos we also introduced the idea of deriving formulas for $f'(x)$ for a general 
$x$, and considering $f'$ as a new function in its own right. We introduced these ``shortcut formulas:''
\begin{enumerate}
\item[(a)]  If $f(x) = x^n$ for any number $n$, then $f'(x) = n x^{n-1}$.
\item[(b)]  If $f(x) = e^x$, then $f'(x) = e^x$ (yes, the same function!)
\item[(c)]  If $f'(x)$ exists, then so does $(kf)'(x)$ and $(kf)'(x) = k f'(x)$.  
\item[(d)]  If $f'(x)$ and $g'(x)$ both exist, then so does $(f + g)'(x)$, and 
$(f + g)'(x) = f'(x) + g'(x)$.  
\end{enumerate}
\vfill\eject
\noindent
{\it Additional Questions}
\begin{enumerate}
\item[(3)] Verify the formula in (a) for $f(x) = \frac{1}{x^2} = x^{-2}$.  
That is, you want to show $f'(x) = -2 x^{-3}$ using the limit definition of $f'(x)$. 
\item[(4)] Using the shortcut rules above, find the derivatives of the following
functions:
\begin{enumerate}
\item[(a)]  $f(x) = x^5 + 2x^{3/2} + 4x^{-3}$
\item[(b)]  $g(x) = x^{1/2} + 3e^x$
\item[(c)]  $h(x) = e^\pi$ (be careful!)
\end{enumerate}
\end{enumerate}
\end{document}
