Week 8: Functions of several variables (part 3)
Dr Jorge Vit´
oria
School of Engineering and Mathematical Sciences
Department of Mathematics
City, University of London
March 27, 2018
Implicit differentiation in several variables.
Last lecture’s considerations on implicit differentiation can be
extended to more than two variables, where one of these
variables is considered as a function in all other variables. More
generally, let
f
be a function in
n
+ 1 variables
x
1
, x
2
, ..., x
n
, y
.
Consider
y
as a variable in the remaining variables; that is,
y
=
y
(
x
1
, x
2
, .., x
n
). Consider the level set consisting of all points
(
x
1
, x
2
, .., x
n
, y
) satisfying
f
(
x
1
, x
2
, ...x
n
, y
) =
c ,
where
c
is some constant.
Dr Jorge Vit´oria
Mathematics for Economics (Post A level)
In order to calculate the partial derivatives
∂y
∂x
i
, we use the
chain rule together with the observation
∂x
i
∂x
i
= 1
,
∂x
i
∂x
j
= 0
,
for 1
≤
i, j
≤
n
and
i
6
=
j
.
Provided that
∂f
∂y
6
= 0 we get as in the case of two variables that
∂y
∂x
i
=

∂f
∂x
i
∂f
∂y
Dr Jorge Vit´oria
Mathematics for Economics (Post A level)
Example.
Consider the function
f
(
x
1
, x
2
, y
) =
x
1

2
x
2

3
y
+
y
2
and the
level curve
f
(
x
1
, x
2
, y
) =

2.
∂y
∂x
1
=

1

3 + 2
y
=
1
3

2
y
,
∂y
∂x
2
=


2

3 + 2
y
=
2
2
y

3
.
These are defined except if
y
= 3
/
2.
Dr Jorge Vit´oria
Mathematics for Economics (Post A level)
Linear Approximations: one variable
Definition.
The
linear approximation to
f
at
x
=
a
is the
tangent line to the graph of
f
at the point (
a, f
(
a
)).
This tangent line is given by the equation
y
=
f
(
a
) +
f
0
(
a
)(
x

a
)
When
x
close to
a
this approximates the value of the function
f
(
x
).
For examples of use in EMEA see section 7.4.
Dr Jorge Vit´oria
Mathematics for Economics (Post A level)
If we set
b
=
f
(
a
), then the tangent line equation can be
rewritten in the form
y

b
=
f
0
(
a
)(
x

a
)
.
Since
y

b
approximates the difference in value of the function
f
as
x
varies around
a
, one finds in the literature the notation
dy
=
f
0
(
a
)
dx
, or also Δ
y
=
f
0
(
a
)Δ
x
.
Dr Jorge Vit´oria
Mathematics for Economics (Post A level)
Example.
Suppose
C
(
x
) is the cost of produce
x
units. Suppose that
C
0
(50) = 2. This information can be interpreted as follows:
Producing 51 units rather than 50 will cost approximately and
extra 2; that is,
C
(51)
≈
C
(50) + 2. This is useful for evaluating
the marginal cost of a product.
Dr Jorge Vit´oria
Mathematics for Economics (Post A level)
Linear approximation: two variables
A function
z
=
f
(
x, y
) in two variables can be represented as a
surface in the 3dimensional space. If
f
is linear, that is of the
form
z
=
ax
+
by
+
c
for some constants
a
,
b
,
c
, then this surface is a plane.
Dr Jorge Vit´oria
Mathematics for Economics (Post A level)