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ECON 302
Math Review
Ali Toossi
Suppose Y = f (X). Define ΔY = f (X + ΔX) – f (X), where ΔY and ΔX denote changes in
Y and X respectively.
Slope at X
0
= Derivative of the function at X
0
:
X
Y
dX
dY
X
∆
∆
=
→
∆
0
lim
Example1:
Consider a linear function Y = f (X) = a + m X. Find slope at X = X
0
.
Solution
:
ΔY = f (X
0
+ ΔX) – f (X
0
) = [a + m (X
0
+ ΔX)] – [a + m X
0
] = m ΔX
Slope at X
0
= Derivative of the function at X
0
=
X
Y
dX
dY
X
∆
∆
=
→
∆
0
lim
=
0
lim
→
∆
X
(m
∆
X) /
∆
X =
0
lim
→
∆
X
m = m.
So slope is constant and does not depend on X
0
.
Example2:
Consider the function Y = f (X) = a X
2
. Find slope at X = X
0
.
Solution
:
ΔY = f (X
0
+ ΔX) – f (X
0
) = a (X
0
+ ΔX)
2
– [a X
0
2
] = (a X
0
2
+ 2a X
0
ΔX + a ΔX
2
)  a X
0
2
= 2a X
0
ΔX + a ΔX
2
∆
Y/
∆
X = (2a X
0
ΔX + a ΔX
2
) /
∆
X = 2a X
0
+ a ΔX
Slope
:
X
Y
X
∆
∆
→
∆
0
lim
=
0
lim
→
∆
X
(
2a X
0
+ a ΔX) = 2a X
0
.
So slope is not constant and depends on X
0
. For example if
X
0
= 4, slope is
8a
, but if X
0
= 0.5, slope is
a
.
Rules of Differentiation (i.e. rules of finding the derivatives of functions):
ConstantFunction Rule
0
=
⇒
=
dX
dY
K
Y
where k is a constant
Product Rule
dx
x
dg
x
f
dx
x
df
x
g
dX
dY
x
g
x
f
Y
)
(
)
(
)
(
)
(
)
(
)
(
+
=
⇒
=
Power Function Rule
1

=
⇒
=
n
n
ncx
dX
dY
cx
Y
, where
c
and
n
are constants.
Quotient Rule
)
(
)
(
)
(
)
(
)
(
)
(
)
(
2
x
g
dx
x
dg
x
f
dx
x
df
x
g
dX
dY
x
g
x
f
Y

=
⇒
=
SumDifference Rule
dx
x
dg
dx
x
df
dX
dY
x
g
x
f
Y
)
(
)
(
)
(
)
(
±
=
⇒
±
=
Chain Rule
Suppose that we have a function Z = f(Y) where
Y = g(X), then:
dx
dy
dy
dz
dx
dz
=
1
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View Full Document Partial Derivatives:
Suppose Y = f(K, L), i.e. Y depends on K & L both. We are interested in knowing what
happens to Y if we change K by an infinitesimally small amount. That is, suppose we
want to know the derivative of Y with respect to K. What about L? Suppose we are
prepared to assume that changing K by a little bit does not cause L to change
at all
. In
such a case we can find the
partial
derivative of Y with respect to K.
The partial
derivative is defined as the effect on Y of a very small change in K
holding L
constant
.
The partial derivative of Y with respect to K is written as:
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This note was uploaded on 02/14/2011 for the course ECON 302 taught by Professor Avrinrad during the Spring '09 term at University of Illinois, Urbana Champaign.
 Spring '09
 AVRINRAD
 Economics

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