Math 137 notes - FALL 2019: October 14 - November 8
Ian Payne
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1

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October 21
The Derivative Function
Definition.
Let
f
(
x
) be a function and
I
be an open interval. We say that
f
(
x
)
is differentiable on
I
if it is differentiable at every point on
I
. In this case, the
function
f
0
(
x
) is defined on
I
by
f
0
(
a
) =
f
0
(
a
) for each
a
.
1
Note that this implies
f
(
x
) is defined on all of
I
, and from a Theorem from
last class, it implies it is continuous on
I
.
There are similar ways of extending
the notion of differentiability to closed intervals by taking some care with the end
points, but I’m not going to do this. To be differentiable at a point, you really
require the function be defined on an open interval containing it.
Definition.
If
f
(
x
) is differentiable on all of
R
, we say that
f
(
x
) is differentiable.
I don’t think I mentioned this earlier, but the same goes for the word “contin-
uous”.
Notation.
This notation is due to Leibniz: we say
d
dx
f
(
x
) or
df
dx
means the same
as
f
0
(
x
).
The creature
d
dx
is called an “operator”, which is a fancy word for “function”,
but we use a different word because it usually takes functions as its input. Indeed,
you can think of
d
dx
as a function which takes a function (whose variable is called
x
) as input, and outputs another function (the derivative) whose input variable is
x
. If the function is called
f
(
t
), the operator would be written
d
dt
.
It is important
to note that this is not a fraction. The notation is meant to remind us
that derivatives are slopes, which are calculated as the limit of fractions.
There are also ways, in particular when you get to integration, where
df
dx
can be treated rather like a fraction.