Section 3.1: Derivatives of Polynomials and Exponential Functions
Instructor: Ms. Hoa Nguyen
([email protected])
Power Rule
If
f
(
x
) =
x
n
(
n
is any real number) then
f
(
x
) =
nx
n

1
.
Example
:
•
f
(
x
) =
x
1000
⇒
f
(
x
) =
•
f
(
t
) =
t
3
⇒
f
(
t
) =
•
f
(
x
) =
x
⇒
f
(
x
) =
•
f
(
x
) =
c
(where
c
is a constant)
⇒
f
(
x
) =
•
f
(
r
) =

1
r
2
⇒
f
(
r
) =
•
f
(
x
) =
√
x
⇒
f
(
x
) =
•
f
(
x
) =
3
√
x
2
⇒
f
(
x
) =
•
f
(
x
) =
x
√
x
⇒
f
(
x
) =
Remark
:
f
=
df
dx
(Leibniz notation)
Compute Derivatives of “New” Functions
“New” functions are formed from “old” functions by addition, subtraction, multiplication or division. If we
know the derivatives of “old” functions, we can compute the derivatives of the “new” functions based on the
following rules:
Assume that the derivatives of
f, g
exist (i.e.,
f, g
are differentiable), then
The constant multiple rule
:
d
dx
[
cf
] =
c
df
dx
⇔
(
cf
) =
cf
(
c
is a constant)
The sum or difference rule
:
d
dx
[
f
±
g
] =
df
dx
±
dg
dx
⇔
(
f
±
g
) =
f
±
g
We will learn about
product or quotient rule
on the next section.
Example 5 of Section 3.1
(textbook):
Example 6 of Section 3.1
(textbook):
Example 7 of Section 3.1
(textbook):
Exponential function
If
f
(
x
) =
a
x
(
a >
0
, a
= 1
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 Fall '08
 Noohi
 Calculus, Exponential Function, Polynomials, Derivative, Exponential Functions, Power Rule, Ms. Hoa Nguyen, [email protected]

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