\documentclass{amsart}[11pt]
\usepackage{graphicx}

\addtolength{\textwidth}{1.4in}                          
\oddsidemargin=0in                          
\evensidemargin=0in       

\newtheorem{example}{Example} 
\newtheorem{question}[example]{Question}
\newtheorem{definition}[example]{Definition}
\newtheorem{theorem}[example]{Theorem}
\newtheorem{notation}[example]{Notation}
\newtheorem{fact}[example]{Fact}

\begin{document}

\title{Math 241, Lecture 9}
\maketitle

\section{ The Derivative}

The derivative of a function $f$ at some point $x$ is the slope of the tangent
line to the graph of $f$ at the point $(x, f(x))$.    We can collect these numbers
all together to make a new function, which we also call the derivative and
which we name $f'$.   
We are about to learn many formal rules to find this $f'$, but before we do that
we should play a game to develop some intuition about the derivative.  

\begin{example}
The derivative game: given some graphs of  derivative functions, sketch possible
graphs for the original functions.  
\end{example}


\section{ Formula for instantaneous rate of change - the algebraic viewpoint}  

Last time we found what the derivative must be by looking
at slopes of secant lines which were close to the tangent line.  
One point on the secant line was always $(x, f(x))$.  
The other we chose to be $(x + h, f(x+h))$, where we understood
that $h$ was meant to be small.     The slope of this secant
line would then be $\frac{f(x + h) - f(x)}{x + h  - x} = \frac{f(x + h) - f(x)}{h}$. 

\begin{definition}
The derivative of $f(x)$ with respect to $x$ is the function 
$$f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}.$$  
\end{definition}

We will talk about $\lim$, which means the ``limit'', later.   
For now, we will adopt a working notion that we may compute
this for smaller and smaller $h$; if all of those computations seem
to approach a single number, for now we will call that the limit.  
We will often leave $h$ in the equation ``until the very end'', at which
point we ``let it become very small - so small it is negligible''.  

\begin{example}
Find the derivative of $f(x) = \frac{1}{x}$ at $x = 2$ first and then for
any possible value of $x$.  
\end{example}

\begin{example}
Find the derivative of $f(x) = -\frac{1}{2} x + 3$.   
Is the answer expected?   
\end{example}

\begin{example}
Suppose price index, measuring the aggregate price for a large 
cross-section of household goods measured in thousands of dollars, 
has values $2 - \frac{1}{1+x}$ over two years.  
What is the rate of price increase over two years?    What is the
``instantaneous rate of inflation'' at six months, one year,
and two years?  
\end{example}

\section{ First rules of differentiation}

It would be tedious to compute a difference quotient and take a limit
every time one had to take a derivative,   just as it would be 
tedious to add $57$ to itself $13$ times in order to compute $57 \times 13$.  
Our next class times will be devoted to learning efficient ways to 
compute derivatives.    Just like doing multiplication, there are both
some basic cases to memorize,   and some rules to reduce complicated
situations to simpler ones.  

First, it is convenient to use a different name for the derivative.  

\begin{notation}
The derivative of $f(x)$ is sometimes denoted by $\frac{d}{dx} f(x)$.   
\end{notation}

Next, we state a rule which relates for example the derivatives of $x^3$
and $-53 x^3$.  

\begin{theorem}
If $g(x) = c f(x)$ then $\frac{d}{dx} g(x) = c \frac{d}{dx}$.  
\end{theorem}

\begin{example}
If the derivative of $f(x) = \sqrt{x}$ is $-\frac{1}{2 \sqrt{x}}$, what is the
derivative of $g(x) = 10 \sqrt{x}$?  
\end{example}

This rule can be deduced from the definition of the 
derivative, and some rules for limits.  
Our second rule, which may be similarly deduced, 
deals with addition of functions.  

\begin{theorem}
$\frac{d}{dx} \{ f(x) + g(x) \} = \frac{d}{dx} f(x) + \frac{d}{dx} g(x)$.  
\end{theorem}

\begin{example}
What is the derivative of $3 x^2 + \frac{1}{x}$? 
\end{example}


 

Our next rule is one of the most famous, and one
of which we have already seen many examples.  

\begin{theorem}
$\frac{d}{dx}  x^n = n x^{n-1}$.  
\end{theorem}

This rule works even when $n$ is a fraction, or negative, or zero.  
We may check it for all of the cases which we have seen
so far, as well as the following.  

\begin{example}
\begin{itemize}
\item Find the derivative of $x^{10000}$ and of $\frac{1}{x^3}$.  
\item Find the derivative of $2 x^{\frac{7}{4}} - \sqrt{\frac{5}{x}}$.  
\item Find the derivative of $ 3x^{500} - 5.734 + x^{-500}$.  
\end{itemize}
\end{example}




\end{document}
 