Here is a picture of the graph of a function y = f
(x)
and a line tangent to the graph at
the point
(x0,y0)
.
PICT
Thus y_{0} = f
(x0)
. Suppose m is the slope of this line. Then from algebra, the equation
of the line is
y = y0 + m (x− x0) = f (x0)+ m (x − x0)
The problem is to determine what m should be so that the above picture is in some sense
correct. Imagine what would happen if you changed m. Then you would get a line which
would not be tangent to the curve at this point. It might intersect the curve in more than
one point for example.
Consider the function g
(x)
=
|f (x)− (f (x )+ m (x− x ))|
0 0
. From the above picture,
this ought to look like the curved part of the following graph in which I have also given
the graph of y = ε
|x− x |
0
where ε is an arbitrary positive number, possibly
small.
PICT
From the picture, you see that if
|x− x0|
is small enough, then
|g (x)|
≤ ε
|x− x0|
.
That is,
|f (x)− (f (x )+ m (x− x ))|
-----------0----------0-- < ε
|x− x0|
In other words, m should be chosen in such a way that
|f (x)−-(f (x0)+-m-(x−-x0))|
lx→imx0 |x− x0| = 0
this from the definition of the limit. Thus, rewriting the above,
| |
||f (x)−-f-(x0) ||
lxi→mx0| x− x0 − m| = 0
This motivates the following definition of the derivative.
Definition 5.1.1Let f be a function defined near x_{0}. Then f is said to bedifferentiable at x_{0}if there is a number m, called the derivative such that
lim |f (x)−-(f (x0)+-m-(x−-x0))|= 0
x→x0 |x− x0|
equivalently
lim f (x)−-f (x0)-= m
x→x0 x− x0
This number may be denoted as f^{′}
(x0)
or
dy
dx
(x0)
or
df
dx
(x0)
or y^{′}orẏor y_{
x}or Df
(x)
.If you let h = x − x_{0}, this is equivalent to
lim f (x0-+h-)−-f (x0)= m ≡ f ′(x )
h→0 h 0
In this book, the last formulation of the definition will be used most of the time.
f(x)− f(x )
---x−-x0-0
or
f(x +h)− f(x )
--0--h---0-
are called difference quotients.
Geometrically, f^{′}
(x0)
measures the slope of the tangent line. However, it is much
better to think of it as a mapping which takes numbers to numbers as follows.
x → f^{′}
(x0)
x. The distinction is more important later when the derivative of a function
of many variables is discussed.
Example 5.1.2Let f
(x)
= x^{n}where n is a positive integer. Find f^{′}
(x)
.
First assume n is a positive integer. Set up the difference quotient and use the
binomial theorem, Theorem 1.6.1.
∑n ( n ) n− k k n
f (x + h)− f (x) (x + h)n − xn k=0 k x h − x
------h------- = -----h------= -----------h-----------
∑ ( ) ∑ ( )
nk=1 n xn−khk nxn− 1h + nk=2 n xn−khk
= -------k----------= ----------------k----------
h ( ) h
h2∑n n xn−khk−2
= nxn −1 +----k=2---k-----------
( )h
n−1 ∑n n n−k k−2
= nx + h k x h
k=2
It follows that
lim f-(x-+-h)−-f (x) = nxn−1
h→0 h
because the second term on the right is something multiplied by h.
Example 5.1.3Let f
(x)
=
|x|
. Show f^{′}
(0)
does not exist.
If it did exist, then the limit of the difference quotient from the right and from the left
would need to coincide. However,
lim f (0+-h)-− f-(0) = lim h-= 1
h→0+ h h→0+ h
f (0+-h)-− f-(0) − h
hli→m0− h = hli→m0− h = − 1
The following diagram shows how continuity at a point and differentiability there are
related.
PICT
Theorem 5.1.4If f is defined near x and f is differentiable at x then fis continuous at x.
Proof: Suppose lim_{n→∞}x_{n} = x. Does it follow that lim_{n→∞}f
(xn)
= f
(x )
?
Let
{
|f(xn)−f(x)| if xn ⁄= x
hn(x) ≡ 0 i|xfn x−x|= x
n
Both Bolzano and Weierstrass gave examples of functions which are continuous at
every point yet differentiable at no point, but you can easily see the example of y = |x|
which is continuous at 0 but not differentiable there.