14. Derivatives
July 26, 2019
Derivative of a function
Definition
Let
I
⊂
R
be an interval, and let
f
:
I
→
R
be a
function.
1. The function
f
is
differentiable at
c
∈
I
if the limit
lim
x
→
c
f
(
x
)
-
f
(
c
)
x
-
c
(1)
exists.
2. The limit is called the
derivative of
f
at c
and is denoted by
f
0
(
c
).
3. If
f
is differentiable at each point
x
in a set
S
⊆
I
, then
f
is
differentiable on S
, and
the function
f
0
:
S
→
R
is called the
derivative
of
f
.
[Examples and remark]
1. Another version of the limit (1):
lim
h
→
0
f
(
c
+
h
)
-
f
(
c
)
h
.
2. Let
f
(
x
) =
k
be constant.
For any
c
∈
R
,
f
(
c
+
h
)
-
f
(
c
)
h
=
k
-
k
h
=
0
h
= 0
.
Then
f
0
(
c
) = lim
h
→
0
f
(
c
+
h
)
-
f
(
c
)
h
= 0.
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117
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