92.131
Lecture 13
1 of 14
Ronald Brent © 2008 All rights reserved.
The Derivative as a Function
Let’s extend the definition of the derivative at a point to a derivative function.
h
x
f
h
x
f
x
f
h
)
(
)
(
lim
)
(
0
−
+
=
′
→
Notes:
•
In essence we are simply replacing
a
with
x
creating a formula for the
rate function associated with
)
(
x
f
.
•
We can’t just let
0
=
h
, some algebraic manipulation or trick is needed.
•
It may be impossible to find the derivative, especially if
)
(
x
f
has a
very complicated formula.
In this case a numerical method is needed.
•
The domain of
)
(
x
f
′
is the set of inputs
x
for which the limit exists.
In
many cases this is the full domain of
)
(
x
f
.
However in some cases it
isn’t, for example the function
x
x
f
=
)
(
has no derivative at
0
=
x
.

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