1 Derivatives: The Five Basic Rules
1.1 Nonlinear Functions
The term derivative means ”slope” or rate of change. The
f
ve rules we are about
to learn allow us to
f
nd the slope of about 90% of functions used in economics,
business, and social sciences.
Suppose we have a function
y
=
f
(
x
)
(1)
where
f
(
x
)
is a non linear function. For example:
1
y
=
x
2
2
y
=3
√
x
x
1
/
2
3
y
=
ax
+
bx
2
+
c
(2)
Each equation is illustrated in Figure 1.
1.2 The Derivative
Given the general function
y
=
f
(
x
)
the derivative of
y
is denoted as
dy
dx
=
f
0
(
x
)(
=
y
0
)
The symbol
dy
dx
is an abbreviation for ”the change in
y
(
dy
) FROM a change in
x
(
dx
)”; or the ”rise over the run”. In other words, the slope.
1.3 The Five Rules
1.3.1
The Constant Rule
Given
y
=
f
(
x
)=
c,
where
c
is an arbitrary constant, then
dy
dx
=
f
0
(
x
)=0
(3)
1.3.2
Power Function Rule
Suppose
y
=
ax
n
(4)
where
a
and
n
are any two constants. The power function rule states that the
slope of the function is given by
dy
dx
=
f
0
(
x
anx
n
−
1
(5)
1
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y
x
x
Y = x
2
Y = x
1/2
x
Y = ax
2
+ bx +c
Examples of
common nonlinear
functions
Figure 1:
2
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 Fall '11
 WendyWu
 Economics, Calculus, Derivative, dx, dy

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