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The derivative at a point and the derivative as a
function.
The derivative of
f
at
a
is denoted by
f
′
(
a
)
and is defined as
lim
h
→
0
f
(
a
+
h
)
−
f
(
a
)
h
if this limit exists.
Alternatively it could be defined as
lim
x
→
a
f
(
x
)
−
f
(
a
)
x
−
a
.
Intuitively this expression is the limit of the slope of a
secant line from
(
a
,
f
(
a
)
)
to
(
x
,
f
(
x
)
)
(or to
(
a
+
h
,
f
(
a
+
h
)
)
with the first definition).
If
y
=
f
(
x
)
,
and we define
Δ
y
=
f
(
x
2
)
−
f
(
x
1
)
and
Δ
x
=
x
2
−
x
1
,
then the instantaneous rate of change of
f
at
x
1
is
f
′
(
x
1
)
=
lim
x
2
→
x
1
Δ
y
Δ
x
.
The derivative of
f
is denoted by
f
′
(
x
)
and is defined as
lim
h
→
0
f
(
x
+
h
)
−
f
(
x
)
h
(also called the Newton quotient).
Alternatively,
f
′
(
x
)
is denoted by
ⅆ
ⅆ
x
(
f
(
x
)
)
in the Leibniz notation.
If
y
=
f
(
x
)
,
then we also write
y
′
=
ⅆ
y
ⅆ
x
. This is not a fraction of two quantities, but rather a single expression
representing the derivative.
In this notation
f
′
(
a
)
is denoted by
ⅆ
y
ⅆ
x
|
x
=
a
A function
f
is differentiable at a value
c
if
f
′
(
c
)
exists.
A function
f
is differentiable on an open interval
(
a
,
b
)
if it is differentiable at each value in
(
a
,
b
)
.