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\begin{document}


\title{On-line Math 21}


\email{david.johnson@lehigh.edu}


\urladdr{http://www.lehigh.edu/\textasciitilde{}dlj0}

\maketitle

\subsection{The definition of the derivative}


\subsubsection{Tangents and Velocity}

What is the ``derivative''? The easy answer is that the derivative of a  function
\( f(x) \) at a point \( x \) is the slope of the curve \( y=f(x) \) at the
 point \( (x,f(x)) \) corresponding to that value of \( x \). But, talking
about the slope of a curve doesn't give any idea of the uses of the concept.
Let's start with the idea of slope, however, since that is the easiest. 

\begin{example}
\textbf{Slope}
\end{example}
Consider the function \( f(x)=x^{2} \). What do we mean by the \textit{slope}
of the curve \( y=x^{2} \) at a point? Let's take  the point with \( x \)-value
2, \( (2,4) \). If the curve were a line, then its slope would obviously be
the slope as is defined for a line, the rise over the run, or the change in
\( y \) over the change in \( x \), \[
\frac{\Delta y}{\Delta x}.\]
 We often use the \( \Delta  \) to indicate the change of a quantity or variable. 

But, this curve is not a line, and so we have to think a bit more subtly. We
can easily think about the slope of the line between two points on the curve.
This is called a \textit{secant line}. I don't know why. Take one of the two
 points to be \( (2,4) \), and the other to be nearby, say \( (2+h,(2+h)^{2}) \).
The  slope of the line connecting those points will be  \[
slope=\frac{(2+h)^{2}-4}{(2+h)-2}=\frac{(4+4h+h^{2})-4}{h}=\frac{4h+h^{2}}{h}=4+h.\]


So, in this case it's easy to see what the slope of the curve will be at \( (2,4) \),
you simply take \( h \) smaller and smaller, getting secant lines that are
closer  and closer to the line tangent to the curve at \( (2,4) \). The \textit{limit}
as \( h\rightarrow 0 \) gives the slope of the curve: \[
\mathrm{Slope}\, \mathrm{of}\, \mathrm{the}\, \mathrm{curve}\, =\lim _{h\rightarrow 0}\frac{(2+h)^{2}-4}{(2+h)-2}=4.\]


\begin{example}
\textbf{Velocity} 
\end{example}
High-school physics texts usually state that \textit{velocity = distance \( \div  \)
time}, or some such. That's right as far as it goes, which really assumes that
the speed (or velocity) is constant. If the speed isn't constant, which really
 is usually the case, the notion needs to be extended. First, let \( D_{1} \)
and  \( D_{2} \) be the positions of some moving object at times \( t_{1} \)
and \( t_{2} \). The \textit{average velocity} over that time interval is the
change in position  divided by the change in time, \[
v_{av}=\frac{D_{2}-D_{1}}{t_{2}-t_{1}}.\]


Just as before, we can find, in a given example, the limit of the average  velocity
as the time interval decreases. That is the instantaneous velocity: \[
v=\lim _{t_{2}\rightarrow t_{1}}\frac{D_{2}-D_{1}}{t_{2}-t_{1}}.\]


We'll go over a specific problem in class, where the position at time \( t \)
is \( s(t)=-16t^{2}+64t+2 \) feet above the ground. 


\subsubsection{Definition of the derivative}

The derivative of a function \( f(x) \) at a particular value \( x=a \) has
the  following formal definition. All it is is a general statement of the idea
of  finding the slope of the curve at \( (a,f(a)) \). 

\begin{defn}
Let \( f \) be a function which is defined on at least some open interval that
contains the number \( x \). Then, the \textit{derivative} of  \( f \) at
\( x \) is the limit \[
f'(a):=\lim _{h\rightarrow 0}\frac{f(a+h)-f(a)}{h}.\]
The function \( f \) is \textit{differentiable} at \( a \) if this limit exists.
If \( f \) is differentiable at all \( x \) in its domain, then \( f \) is
a \textit{differentiable function}.  

We usually don't worry about just the slope at one value, but want a formula
to find the slope at any \( x \) in the domain of \( f \), so that slope,
as as function of \( x \), is written as  \[
f'(x):=\lim _{h\rightarrow 0}\frac{f(x+h)-f(x)}{h}.\]

\end{defn}
The fraction \[
\frac{f(x+h)-f(x)}{h},\]
 for any given \( h \), is called a \textit{difference quotient} for the function
\( f \).  

\begin{example}
Find \( f'(2) \) for the function \( f(x)=x^{3} \). 
\end{example}

\subsubsection*{Solution }

\begin{example}
Find \( f'(x) \) for the function \( f(x)=\sqrt{x} \). 
\end{example}

\subsubsection*{Solution }

\begin{example}
Find the derivative of the function \( f(x)=2^{x} \).
\end{example}

\subsubsection*{Solution}


\subsection*{The derivative as a function}

\begin{defn}
Let \( f \) be a function which is defined on at least some open interval that
contains the number \( x \). Then, the \textit{derivative} of  \( f \) at
\( x \) is the limit \[
f'(x):=\lim _{h\rightarrow 0}\frac{f(x+h)-f(x)}{h}.\]
The function \( f \) is \textit{differentiable} at \( x \) if this limit exists.
If \( f \) is differentiable at all \( x \) in its domain, then \( f \) is
a \textit{differentiable function}.  
\end{defn}

\subsection*{More on derivatives }

Since the derivative is the limit as the change in \( x \) goes to 0 of the
ratio of the change in \( f \) over the range in \( x \), that is: \[
f'(x)=\lim _{\Delta x\rightarrow 0}\frac{\Delta f}{\Delta x}=\lim _{\Delta x\rightarrow 0}\frac{f(x+\Delta x)-f(x)}{\Delta x},\]


we often denote \( f'(x) \) as\footnote{%
The \( df/dx \) notation was developed by Leibniz, who was one of the first
people to develop calculus. The other principal inventor of calculus was Issac
Newton, who used the \( f'(x) \) notation. Leibniz' notation seems very ornate,
appropriately very German. The comparison would have been better had Newton
been French, since the \( f'(x) \) notation almost seems French. But no, he
was English. Actually, it seems that Newton preferred to denote the derivative
with a dot over the function, but this standard notation has been attributed
to him.

The links in the paragraphs above are to biographies of the two inventors of
calculus, including something of the dispute about who invented what first.
} \[
f'(x)=\frac{df}{dx}.\]
 

There are some other common terminologies for the derivative. If \( y=f(x) \)
 describes a function, then all of these mean the same thing, the derivative
of the function: \[
f'(x)=\frac{df}{dx}=\frac{d}{dx}(f(x))=D(f)(x)=D_{x}(f)(x)=f_{x}(x)=\frac{dy}{dx}=D(y).\]

\end{document}
