DERIVATIVE AS A FUNCTION
If we replace
a
by
x
in the definition of the derivative of a function
f
at a number
a
, we can get
f
0
(
x
) = lim
h
→
0
f
(
x
+
h
)

f
(
x
)
h
.
So we can define a function that gives us the slope of the tangent line at each point,
and we call it the derivative of
f
.
Example 1.
Find The derivative of the function
f
(
x
) =
1
x
.
Solution
f
0
(
x
)
=
lim
h
→
0
f
(
x
+
h
)

f
(
x
)
h
=
lim
h
→
0
1
x
+
h

1
x
h
= lim
h
→
0

h
(
x
+
h
)
x
h
= lim
h
→
0

1
(
x
+
h
)
x
=

1
x
2
.
Let’s look at the graph of
f
(
x
) =
1
x
. If we consider the slope of the tangent line,
we can see that it is negative when
x >
0, but slope becomes closer to 0 as
x
→ ∞
.
By looking at the graph, we can sketch the graph of the derivative, and vise versa.
Example 2.
The graph of a function
f
is given. Use this to sketch the graph
of its derivative
f
0
.
Historically, mathematicians used lots of different notations for the derivative.
For example,
f
0
(
x
) =
y
0
=
dy
dx
=
lim
Δ
x
→
0
Δ
y
Δ
x
=
df
dx
=
lim
Δ
x
→
0
Δ
f
Δ
x
=
d
dx
f
(
x
) =
Df
(
x
)
and
f
0
(
a
) =
y
0
x
=
a
=
dy
dx
x
=
a
=
df
dx
x
=
a
Figure 1.
f
(
x
)
1
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 Fall '09
 BenedictGross
 Calculus, Derivative, Slope, lim, Continuous function, lim h0

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