Differentiability
The Difference Quotient and the Derivative
Today we will study what it exactly means for a function to have a derivative. In doing so, we will talk
about a specific rational function, called the difference quotient, and how it it used to define the derivative.
We will discuss functions which are continuous at a point but do not have a derivative at that point, and
we will talk about functions defined on a closed interval and how being defined on a closed interval impacts
the search for maxima and minima.
Let f(x) be any real valued function, and let abe a value of xat which f(x) is defined. We will define a
new function δ(x) by
δ(x) = f(x)−f(a)
x−a.
The function δ(x) is called the difference quotient of f(x) at x=a. It is defined everywhere f(x) is defined
except for x=a:
Dom(δ) = {x∈R:x∈Dom(f), x 6=a}.
You may recognize this difference quotient as being very much like the slope formulae: indeed, we have seen
a formula like this once before. Geometrically, we can represent δ(p) for some value pof xat which δ(x) is
defined as being the slope of the line passing through the graph of f(x) at x=pand at x=a. If you are
reading these notes on your own, you should sketch out the graph of a typical function f(x), pick a value
for aand a value of p, and draw the line passing through the points (a,f (a) and (p, f (p)). The slope of this
line then represents δ(p).
Given that our last lecture dealt with rational functions and how to find their limits at points outside of
their domains, it should seem natural to you that we are going to find the limit of δ(x) as xapproaches a.
Geometrically, we see this limit as being the slope of the line we drew above as pgets closer and closer to a
from either side. As pgets closer and closer, that line begins to look more and more like the tangent line to
f(x) at x=a. Since δ(x) represents the slope of this line, the limit of δ(x) as xapproaches afrom either
side, if it exists, will equal the slope of this tangent line. We have a name for the slope of the tangent line
to f(x) at x=a: the derivative. Therefore we write:
df
dx(a) = lim
x→a
f(x)−f(a)
x−a.
This is the definition of the derivative of f(x) at x=a. It is the limit of a rational function, the difference
quotient of f(x) at x=a. We say that f(x) is differentiable at x=aif this limit exists. If this limit does
not exist, we say that ais a point of non-differentiability for f(x). If f(x) is differentiable at every point
in its domain, we say that f(x) is a differentiable function on its domain. So far, every function we have
studied in this class, with the exception of the piecewise continuous functions, is differentiable everywhere
on its domain: this includes polynomial functions, the trigonometric functions (we emphasize that we are
talking about having a derivative where they are defined), the exponential functions, the power functions,
and all stretches and shifts of these functions.
Now that we have a limit definition for the derivative and a reasonable idea of how to take a limit, we are
going to discuss the derivatives of some basic functions again in this context. All of the formulae you have
learned in this class for the derivatives of functions can be derived from the limit definition of the derivative.
For example, consider a constant function f(x) = c. Plugging this formula into the limit definition for the
derivative, we get that
df
dx(a) = lim
x→a
f(x)−f(a)
x−a= lim
x→a
c−c
x−a= lim
x→a
0
x−a.
Now, the function 0
x−ais the zero function everywhere except at x=a, where it is undefined. Therefore
that last limit is equal to 0. Thus we have shown that
df
dx(a) = 0,
which is precisely the formula that we have been using throughout this course.
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