ECONOMICS 100A
MATHEMATICAL HANDOUT
Fall 2007
Professor Mark Machina
A.
CALCULUS REVIEW
Derivatives, Partial Derivatives and the Chain Rule
*
You should already know what a derivative is. We’ll use the expressions ƒ
6
(
x
) or
d
ƒ(
x
)/
dx
for the
derivative of the function ƒ(
x
). To indicate the derivative of ƒ(
x
) evaluated at the point
x
=
x
*,
we’ll use the expressions ƒ
6
(
x
*) or
d
ƒ(
x
*)/
dx
.
When we have a function of more than one variable, we can consider its derivatives with respect
to each of the variables, that is, each of its
partial derivatives
. We use the expressions:
∂
ƒ(
x
1
,
x
2
)/
∂
x
1
and
ƒ
1
(
x
1
,
x
2
)
interchangeably to denote the partial derivative of ƒ(
x
1
,
x
2
) with respect to its first argument (that
is, with respect to
x
1
). To calculate this, just hold
x
2
fixed (treat it as a constant) so that ƒ(
x
1
,
x
2
)
may be thought of as a function of
x
1
alone, and differentiate it with respect to
x
1
. The notation
for partial derivatives with respect to
x
2
(or in the general case, with respect to
x
i
) is analogous.
For example, if ƒ(
x
1
,
x
2
) =
x
1
2
⋅
x
2
+ 3
x
1
, we have:
∂
ƒ(
x
1
,
x
2
)/
∂
x
1
=
2
x
1
·
x
2
+ 3
and
∂
ƒ(
x
1
,
x
2
)/
∂
x
2
=
x
1
2
The
normal vector
of a function ƒ(
x
1
,...,
x
n
) at the point (
x
1
,...,
x
n
) is just the vector (i.e., ordered
list) of its
n
partial derivatives at that point, that is, the vector:
(
)
1
1
1
1
1
2
1
1
1
2
ƒ(
,...,
)
ƒ(
,...,
)
ƒ(
,...,
)
,
,...,
ƒ (
,...,
),ƒ (
,...,
),...,ƒ (
,...,
)
n
n
n
n
n
n
n
n
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
∂
∂
∂
⎛
⎞
=
⎜
⎟
∂
∂
∂
⎝
⎠
Normal vectors play a key role in the conditions for unconstrained and constrained optimization.
The
chain rule
gives the derivative for a “function of a function.” Thus, if ƒ(
x
)
≡
g
(
h
(
x
)), then
ƒ
6
(
x
)
=
g
6
(
h
(
x
))
⋅
h
6
(
x
)
The chain rule also applies to taking partial derivatives. For example, if ƒ(
x
1
,
x
2
)
≡
g
(
h
(
x
1
,
x
2
)) then
1
2
1
2
1
2
1
1
ƒ(
,
)
(
,
)
( (
,
))
x
x
h x
x
g h x
x
x
x
∂
∂
′
=
⋅
∂
∂
Similarly, if ƒ(
x
1
,
x
2
)
≡
g
(
h
(
x
1
,
x
2
),
k
(
x
1
,
x
2
)) then:
1
2
1
2
1
2
1
1
2
1
2
2
1
2
1
2
1
1
1
ƒ(
,
)
(
,
)
(
,
)
( (
,
), (
,
))
( (
,
), (
,
))
x
x
h x
x
k x
x
g
h x
x
k x
x
g
h x
x
k x
x
x
x
x
∂
∂
∂
=
⋅
+
⋅
∂
∂
∂
The
second derivative
of the function ƒ(
x
) is written:
ƒ
Î
(
x
)
or
d
2
ƒ(
x
)/
dx
2
and it is obtained by differentiating the function ƒ(
x
) twice with respect to
x
(if you want the
value of ƒ
Î
(
⋅
) at some point
x
*, don’t substitute in
x
* until
after
you’ve differentiated twice).
*
If the material in this section is not
already
familiar to you, you may have trouble on both midterms and the final.

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